Addressing health disparities in Hispanic breast cancer: accurate and inexpensive sequencing of BRCA1 and BRCA2 Background: Germline mutations in the BRCA1 and BRCA2 genes account for 20-25 % of inherited breast cancers and about 10 % of all breast cancer cases. Detection of BRCA mutation carriers can lead to therapeutic interventions such as mastectomy, oophorectomy, hormonal prevention therapy, improved screening, and targeted therapies such as PARP-inhibition. We estimate that African Americans and Hispanics are 4-5 times less likely to receive BRCA screening, despite having similar mutation frequencies as non-Jewish Caucasians, who have higher breast cancer mortality. To begin addressing this health disparity, we initiated a nationwide trial of BRCA testing of Latin American women with breast cancer. Patients were recruited through community organizations, clinics, public events, and by mail and Internet. Subjects completed the consent process and questionnaire, and provided a saliva sample by mail or in person. DNA from 120 subjects was used to sequence the entirety of BRCA1 and BRCA2 coding regions and splice sites, and validate pathogenic mutations, with a total material cost of $85/subject. Subjects ranged in age from 23 to 81 years (mean age, 51 years), 6 % had bilateral disease, 57 % were ER/PR+, 23 % HER2+, and 17 % had triple-negative disease. Rates of mortality, triple-negative disease, and BRCA1 and BRCA2 mutational and allelic diversity are all higher in African American and Hispanic populations. Although these populations could benefit significantly from genetic testing and screening, the combination of lower average income; lower insurance coverage; reduced knowledge of the benefits of testing; and mistrust of medical and government agencies, have led to a large disparity in participation [42] [43] [44] . From 64,717 women in the Myriad database, this under-participation is 3.9-4.8 times lower in these two populations, which together account for 29 % of the US population [35] . A similarly low rate of participation from the first 10,000 women tested at Myriad was noted by Forman and Hall [43] . To begin to address these issues in Hispanic populations, we designed a study with a number of potential advantages: a fully de-linked sample not requiring extensive counseling regarding BRCA testing; collection of saliva that can be completed in the home, community clinics or public events; bilingual study materials; and full protection of confidentiality.. Our recruitment success was very modest from online publicity (Facebook, clinicaltrials.gov), and few minority patients participate in public fund-raising events, such Fig. 2 Selected BRCA2 mutations. The Ion Torrent data displayed in IGV [51] is shown for the newly described 6005delT mutation in the left panel, a. The display shows individual forward (F) sequence reads in red and reverse (R) reads in blue. The 6005delT mutation can be seen as a gap in the sequence (arrow) in approximately half of the F and R reads -this is consistent with a heterozygous mutation. Sanger sequencing (not shown) confirmed this mutation. b. Both the Ion Torrent (above) and Sanger sequence (displayed in Mutation Explorer, SoftGenetics, below) for the C1787S and G1788D mutations are displayed in the right panel. The co-occurrence of the C1787S and G1788D variants on the same allele can be clearly seen in the Ion Torrent reads, whereas phase cannot be determined from the Sanger traces. Note: BRCA1 is in reverse orientation in the genome, and so IGV display is of the reverse complement. as Avon fundraising walks. However, partnerships with community groups such as Nueva Vida (Baltimore, Richmond) and the Latino Community Development Agency (Oklahoma City) were successful in recruiting a number of Hispanic women of diverse backgrounds. The remainder of the population samples were collected at the Texas Tech University Health Science Hospitals in Lubbock and El Paso, Texas. By sending and receiving saliva kits through the US mail we were able to keep the cost of the collection/shipping and DNA preparation materials to under $20 per subject. Targeted sequencing, performed in batch sizes of 90 samples, has a materials cost of approximately $50, making the total reagent cost under $100, including Sanger validation but excluding labor. This is an important factor if sample sizes in the order of thousands of subjects are to be eventually attained.. We did identify amplicons in BRCA2 that performed poorly in most samples, having <100X average coverage, comprising 261 bp of coding region and 4 splice sites, and 1 splice site of BRCA1 poorly covered. The initial run of 92 samples had three that underperformed and required repetition (3 %); however, these samples performed well on a second run. Our second run of 46 samples allowed the average coverage to increase from 313-361X to 466X. Running 46 samples on a single chip would raise the cost/sample by $5, and may be advisable for clinical testing.. The BRCA1 gene contains multiple mononucleotide repeats, for which it is challenging to accurately detect 1 bp indels; these regions are the most problematic to detect for all current methods [36, 40, 41] . Early versions of the TVC demonstrated relatively higher error rates on homopolymer sequences [40, 41] . This performance has improved with subsequent version of TVC and three studies demonstrate that 70-130X coverage across homopolymer regions is sufficient to get accurate mutation calls in the BRCA1 and BRCA2 genes ( Table 5 , Additional file 1) [42, 43] . Using the 70X standard of Dacheva et al., besides the poor performing exons mentioned above, there are only 2 A5 homopolymer repeats in coding regions (BRCA2, exon 12, 65X average coverage) that are below this threshold in our data. In a small sample set, we show here that the Hi-Q formulation and current TVC analysis settings result in a substantial reduction in 1 bp indel calls, especially in false positive mononucleotide regions. Manual analysis of BAM files in IGV allows most of these variants to be excluded from consideration, although the difference in background between the two enzymes visible in IGV displays are not dramatic. The addition of overlapping amplicons in the most difficult regions could improve this result. Although we chose an amplicon-based method on Ion Torrent, the samples could also be run on multiple platforms or used in a capture-based method [29] . Color Genomics (https://getcolor.com/) now offers a clinical test of 19 breast cancer related genes for $249. While this initial phase of the project was designed to maximize participation and diversity at a modest cost, it has the disadvantage that subjects with mutations could not be retrospectively identified in order to benefit from the testing. A second phase, in which identifiers will be retained and patients counseled about testing, will now be initiated. We expect recruitment to increase as several individuals and groups declined to participate in the unlinked arm of the study. Population growth; advancement in age of the US Hispanic populations; and large family size make it imperative that innovative means be employed to increase participation in clinical and genetic studies. The increasingly recognized involvement of germline mutations in BRCA1 and BRCA2 in diverse cancers, as well as the active design of targeted therapies, further adds to the need to recruit minority subjects.. To date there have been no large nationwide surveys of Latin American with breast cancer. In the report by Hall et al., only the identities of mutations reaching 4 % or greater were made available [35] . The Clinical Cancer Genetics Community Research Network collected samples and data from 746 patients from 14 clinics concentrated in Southwestern USA, and individual data from centers in Texas, California, and Puerto Rico [46] . Our study adds to the diversity of Hispanic BRCA mutations, and -interestingly -we did not find the most common allele in all mainland US Hispanic studies to date, the BRCA1 185delAG mutation. Of the six mutations we identified, only one, BRCA2 E1308X, has been reported in multiple studies. A much larger study, incorporating all the regional and ethnic diversities of Hispanic populations, will have to be carried out to fully understand mutational diversity, and to aid in the classification of VUS. Clearly, a comprehensive characterization of our samples will require copy number analysis to identify large insertion/deletion mutations. Recurrent large deletions in BRCA1 have been found in both Mexican and Puerto Rican breast cancer patients [19, 20] . Rare germline mutations in other genes have been identified in familial breast cancer, and our samples could be used to scan these genes [32] or complete exomes or genomes.. While we and others have documented that the major next-generation sequencers do an excellent job in identifying single nucleotide variants, they can be deficient in the prediction of insertion and deletion variants, especially in mononucleotide repeats [41] . The BRCA1 and BRCA2 genes have numerous mononucleotide repeat regions, and these areas are rich in known mutations. Thus, a method to increase accuracy of sequencing in repeats would be welcome. We documented that the Hi-Q enzyme can achieve a significantly higher accuracy in sequencing through mononucleotide repeats. When combined with methods or specific assays for the most prevalent large deletions, a high percentage of germline mutations can be identified. A recent study with the same BRCA1/2 panel was tested in a diagnostics laboratory with high accuracy [47] . All known publications using Ion Torrent sequencing instruments to sequence known BRCA1 and BRCA2 mutations are displayed. The number of insertion/ deletion variants (indels), those of 1 base pair (1bp) and those in homopolymer regions (homopol.) are tabulated. In summary, we have succeeded in recruiting and anonymously testing diverse groups of Latin American women with breast cancer in the US, all at a materials cost of less than $100 for samples collection, shipment and sequencing, and confirmation. Using DNA sequencing, we found almost exclusively rare mutations, most of which have been observed in other studies. Expansion of this approach could be a component of a larger effort to improve the application of the benefits of genetic testing to Hispanic American women. Mutations in the BRCA1 and BRCA2 genes result in predisposition to breast and ovarian cancers [1, 2] . In addition, there is increasing evidence that BRCA mutations confer risk for cancers of the prostate, pancreas, stomach and skin [3] [4] [5] ; there is also suggestive evidence for their involvement in esophageal and gastric cancers [6] . In Caucasian and Asian ethnicities, BRCA mutations are associated with basal-type and/or -triple-negative disease (in which the estrogen, progesterone, and HER2 receptors are absent from the tumor); however, the nature of this relationship in other ethnicities is understudied [7] [8] [9] [10] .. The identification of BRCA1/2 carriers is a critical component of breast and ovarian cancer prevention as there are multiple screening, surgical, and chemoprevention strategies that can be employed. Magnetic resonance imaging (MRI) screening is effective in detecting early cancers in BRCA1/2 carriers; mastectomy and oophorectomy can reduce ovarian cancers; and estrogen inhibition can reduce both type of malignancy [11] . However, relatively few minorities have participated in these prevention studies.. Although African American and Hispanic women have a lower incidence of breast cancer, they have a higher mortality. Triple-negative breast cancer is more common in African Americans and Hispanic women, accounting, in part, for this health disparity [12] [13] [14] [15] [16] . A study of over 45,000 women referred for BRCA1 and BRCA2 mutation screening from 2006 to 2008, found that 13-16 % of African Americans, Native Americans and Hispanics possess disease-causing mutations and a high rate of variants of unknown significance [17] . An independent study of 389 African Americans and 425 Hispanic women with incomplete gene sequencing, found that 8-10 % of high-risk subjects have a BRCA1 or BRCA2 gene mutation, and a small Puerto Rican study reported a 52 % mutation rate [18, 19] . Common recurrent mutations, including large deletions in BRCA1 and BRCA2, exist in Hispanic/Latin American communities, which accounted for 35-45 % of the mutation carriers [17] [18] [19] [20] .. The BRCA1 and BRCA2 proteins play a role in the repair of double-stranded breaks in DNA. A synthetic lethal strategy for cancer therapy has been developed using DNA damaging chemotherapy agents to cause single-stranded breaks, combined with poly-ADP ribose polymerase (PARP) inhibitors, to inhibit single-stranded DNA repair. This approach may be particularly effective in BRCA mutation carriers, as the tumor will be unable to repair the double-stranded breaks [21, 22] . Clinical trials of PARP inhibitors demonstrate partial response or stable disease in breast, ovarian and prostate cancer subjects [23] [24] [25] [26] [27] .. Mutations in BRCA1 and BRCA2 have been detected with a variety of techniques including multiple mutation scanning methods and Sanger sequencing (reviewed in [28] ). Next-generation sequencing (NGS) has the benefit of high-throughput, automated sequence analysis, and single strand reads. DNA capture, droplet PCR and multiplex PCR methods of template preparation, and sequencing on 454, Illumina and Ion Torrent platforms have all been employed [29] [30] [31] [32] . A clinical diagnostic laboratory validation of the Ion Torrent platform demonstrated an absence of false negatives and a 10 % false positive rate [33] . With the availability of a three-tube multiplex for the complete BRCA1 and BRCA2 genes we sought to apply this approach to a cohort of Hispanic/Latin American breast cancer patients.

O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O


Greedy colorings of words In the Binary Paint Shop Problem proposed by Epping et al. (2004) [4] one has to find a 0/1-coloring of the letters of a word in which every letter from some alphabet appears twice, such that the two occurrences of each letter are colored differently and the total number of color changes is minimized. Meunier and Sebő (2009) [5] and Amini et al. (2010) [1] gave sufficient conditions for the optimality of a natural greedy algorithm for this problem. Our result is a best possible generalization of their results. We prove that the greedy algorithm optimally colors every suitable subword of a given instance word w if and only if w contains none of the three words (a, b, a, c, c, b), (a, d, d, b, c, c, a, b), and (a, d, d, c, b, c, a, b) as a subword. Furthermore, we relate this to the fact that every member of a family of hypergraphs associated with w is evenly laminar.Proof. Suppose, for a contradiction, that there is a set I(c 1 , c 2 ) in G(w) with a 2 < c 2 < b 2 . If a 2 < c 1 , then, by (1), a 1 < b 1 < a 2 < c 1 < c 2 < b 2 and w contains the subword (a, b, a, c, c, b) , which is a contradiction. If a 1 < c 1 < a 2 , then a 1 < c 1 < a 2 < c 2 , and c 2 < b 2 implies a contradiction to the minimality of b 2 . If c 1 < a 1 , then, by (1), c 1 < b 1 < c 2 < b 2 , and c 1 < a 1 implies a contradiction to the minimality of a 1 . Hence the claim follows.By Claims 1-3, there is a color change between b 1 and a 2 − 1, that is, G(w) contains a set I(c 1 , c 2 ) with b 1 < c 2 < a 2 . We assume that this set is chosen such that c 2 is maximal. By this choice, we obtain the following.Claim 4. There is a unique color change between c 2 and a 2 , which is at position (a 2 − 1) + 1 2 .If c 1 < a 1 , then c 1 < a 1 < c 2 < a 2 , and c 1 < a 1 implies a contradiction to the minimality of a 1 . Hence a 1 < c 1 .Claim 5. If I 1 and I 2 are sets in G(w), then the number of color changes in (I 1 \ I 2 ) ∪ (I 2 \ I 1 ) is even. The following so-called Binary Paint Shop Problem proposed by Epping et al. [4] , has recently received considerable attention:. Given a word w = (w 1 , . . . , w n ) over some alphabet Σ such that each letter from Σ appears twice in w, find a 0/1-coloring f = (f 1 , . . . , f n ) ∈ {0, 1} n of w in which the two occurrences of each letter are colored differently, that is, w i = w j for some i < j implies f i ̸ = f j , such that the number of color changes between consecutive letters is minimized.. A word in which each letter from some alphabet Σ appears either twice or never is called an admissible word. A coloring f of a word w with 0 and 1 is called a 0/1-coloring. A 0/1-coloring f of an admissible word w is called good if the two occurrences of each letter of w are colored differently. If f i ̸ = f i+1 , then we say that there is a color change at position i + 1 2 , that is, there is a color change between i and i + 1.. The Binary Paint Shop Problem is APX-hard [3, 4] but there is a natural greedy algorithm, which works for an admissible word w = (w 1 , . . . , w n ) as follows:. Set g 1 := 0 and for j from 2 up to n, set g j := g j−1 unless there is some i < j with w i = w j and g i = g j−1 . In the latter case, set g j := 1 − g j−1 .. The coloring produced by the greedy algorithm is called the greedy coloring. The following claim is evident.

O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O


Power characteristics of inductive interconnect The power characteristics of an inductive interconnect tines is presented. The matching condition between the driver and the load has an effect on the power consumption since the short-circuit power dissipation may decrease with increasing line width and the dynamic power increases. A madeoff exists between the short-circuit and dynamic power in inductive interconnects. The short-circuit power increases with wider line widths if the line is underdriven. As the short-circuit power is directly proportional to the signal transition time, an analytic solution for the transition time with an error of less than 15% is presented. The solution can be used in wire sizing synthesis techniques to decrease the overall power dissipation. With the decrease in the feahlre size of CMOS circuits, on-chip interconnect now dominates circuit delay and power dissipation. Furthermore, the inductive behavior of the interconnect can not be neglected, particularly in long interconnect lines. The interconnect inductance increases on-chip noise as well as the computational complexity of the design process. Furlhemore, on-chip inductance may affect certain design techniques such as repeater insertion [I] and wiresizing [Z].. Interconnect design has, therefore, become an important issue in high complexity, high speed integrated circuits (IC). Many algorithms have been proposed to determine the optimum wire size that minimizes a cost function such as delay [3] or power. As the inductance becomes important, specific algorithms have been enhanced that consider RLC impedances [4,5].. Previous research in wiresizing does not consider the inductive impedance characteristics, which may lead to non-optimal circuits. For example, the research described in [Z] neglects the impedance matching characteristics between the line driver and the load. In @] wider wires are used to reduce power dissipation. It is shown in lhis paper that this technique may actually increase the total power dissipation. In this paper, the effect of sizing an inductive interconnect line on the sholf-circuit power dissipation of a driven CMOS gateis presented. The changein thepowerdissipation with the interconnect width is characterized in Fig. I , It is shown that the impedance characteristics of an induck Figure 1 . Short-circuit dynamic, rmd total power dissipation as a function of the interconnect line width tive interconnect line are sensitive CO changes in the line width. As the short-circuit power depends directly on the signal transition time, an analytic solution for the signal mansition time at the far end of an inductive interconnect line is presented. This solution can be used in design techniques such as wiresizing to optimize the overall m s i e n t power dissipation.. The paper is organized as follows. In section 2, the power characteristics of an inductive interconnect line are discussed. Furthermore, the effecls of changing the line width on the line impedance characteristics are presented.. In section 3, an analytic solution for the mansition time at the far end of an inductive interconnect is provided. A comparison between the analytical expression and simulation is presented in section 4. Some conclusions are discussed in last section. It is shown in this paper that the short-circuit power within a load gate driven by an inductive interconnect line is dependent upon the impedance characteristics of the line. The matching condition between the driver and the load is shown to have an important effect on the line impedance characteristics. If the line is overdrive", the short-circuit power decreases with increasing line width. When the line exceeds the matched condition, the short-circuit power increases with increasing signal transition time.. An analytic solution of the signal transition time at the far end of an inductive interconnect line is presented. The solution can be used to estimate the uansition time at the far end of an inductive interCoMeCt and exhibits an eror of less than 15%. The analytic solution can be used to both estimate and optimize the power dissipation within high speed CMOS circuits.

O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O B I O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The predecessor-existence problem for k-reversible processes For k ≥ 1, we consider the graph dynamical system known as a kreversible process. In such process, each vertex in the graph has one of two possible states at each discrete time. Each vertex changes its state between the present time and the next if and only if it currently has at least k neighbors in a state different than its own. Given a k-reversible process and a configuration of states assigned to the vertices, the Predecessor Existence problem consists of determining whether this configuration can be generated by the process from another configuration within exactly one time step. We can also extend the problem by asking for the number of configurations from which a given configuration is reachable within one time step. Predecessor Existence can be solved in polynomial time for k = 1, but for k > 1 we show that it is NP-complete. When the graph in question is a tree we show how to solve it in O(n) time and how to count the number of predecessor configurations in O(n 2 ) time. We also solve Predecessor Existence efficiently for the specific case of 2-reversible processes when the maximum degree of a vertex in the graph is no greater than 3. For this case we present an algorithm that runs in O(n) time. Let G be a simple, undirected, finite graph with n vertices and m edges. The set of vertices of G is denoted by V (G) = {v 1 , v 2 , . . . , v n } and its set of edges is denoted by E(G). A k-reversible process on G is an iterative process in which, at each discrete time t, each vertex in G has one of two possible states. A state of a vertex is represented by an integer belonging to the set Q = {−1, +1} and each vertex has its state changed from one time to the next if and only if it currently has at least k neighbors in a state different than its own, where k is a positive integer. Let Y t (v i ) be the state of vertex v i at time t. A configuration of states at time t for the vertices in V (G) is denoted by Y t = (Y t (v 1 ), Y t (v 2 ), . . . , Y t (v n )). The one-step dynamics in a k-reversible process for graph G can be described by a function F . The motivation to study k-reversible processes is related to the analysis of opinion dissemination in social networks. For example, suppose that a network is modeled by a graph, each vertex representing a person and each edge between two vertices indicating that the corresponding persons are friends. Suppose further that state −1 represents disagreement on some issue and that the state +1 means agreement on the same issue. A k-reversible process is an approach to model opinion dissemination when people are strongly influenced by the opinions of their friends and the society they are part of. Notice that in this model we are assuming that all people act in the same manner. A more complex approach could assume, for example, distinct thresholds for each person or thresholds based on one's number of friends.. Note that k-reversible processes are examples of graph dynamical systems; more precisely, of synchronous dynamical systems, which extend the notion of a cellular automaton to arbitrary graph topologies. The study of graph dynamical systems and cellular automata is multidisciplinary and related to several areas, like optics [9] , neural networks [12] , statistical mechanics [1] , as well as opinion [5] and disease dissemination [14] . Also in distributed computing there are several studies regarding models of graph dynamical systems. An example is the model of majority processes in which each vertex changes its state if and only if at least half of its neighbors have a different state [13] . An application of this model is used for maintaining data consistency [16] .. Most of the studies regarding k-reversible processes are related to the Minimum Conversion Set problem in such processes. This problem consists of determining the cardinality of the minimum set of vertices that, if in state +1, lead all vertices in the graph also to state +1 after a finite number of time steps. It has been proved that this problem is NP-hard for k > 1 [6] . There are also several interesting results about this problem in the work by Dreyer [7] , who also presents some important results regarding the periodic behavior of k-reversible processes, as well as upper bounds on the transient length that precedes periodicity. Most of the results presented by Dreyer are based on reductions from the so-called threshold processes, which are broadly studied by Goles and Olivos [11, 10] . Another approach to study the transient and periodic behavior of kreversible processes is the use of a specific energy function that leads to a much more intuitive proof of the maximum period length and also better bounds on the transient length [15] .. A problem that arises in synchronous dynamical systems on graphs is the so-called Predecessor Existence problem, defined as follows. Given one such dynamical system and a configuration of states, the question is whether this configuration can be generated from another configuration in a single time step using the system's update rule. In the affirmative case, such configuration is called a predecessor of the one that was given initially. This problem was studied by Sutner [17] within the context of cellular automata, where configurations lacking a predecessor configuration are known as garden-of-Eden configurations, and was proved to be NP-complete for finite cellular automata. NP-completeness results for related dynamical systems as well as polynomialtime algorithms for some graph classes can also be found in the literature [3] . An extension of the Predecessor Existence problem is to count the number of predecessor configurations. This is also a hard problem and has been proved to be #P-complete [18] .. For k-reversible processes, we address these two problems in this paper. We are interested in determining whether a configuration Y t−1 exists for which Y t = F k G (Y t−1 ). Because only time steps t − 1 and t matter, for simplicity we denote Y t−1 and Y t by Y ′ and Y , respectively. We henceforth denote this special case of Predecessor Existence by Pre(k). We also consider the associated counting problem, #Pre(k), which asks for the number of predecessor configurations. Our results include an NP-completeness proof for the general case of k-reversible processes and polynomial-time algorithms for some particular cases.. The remainder of the paper is organized as follows. In Section 2 we show that Pre(1) is polynomial-time solvable. In Section 3 we provide an NPcompleteness proof of Pre(k) for k > 1. In Section 4 we describe two efficient algorithms for trees, one for solving Pre(k) and the other for solving #Pre(k). In Section 5 we show an efficient algorithm to solve Pre(2) for graphs with maximum degree no greater than 3. Section 6 contains our conclusions.. 2 Polynomial-time solvability of Pre(1). ′ exists then any pair of neighbors u and v for which. Based on this observation, we start by partitioning G into connected subgraphs that are maximal with respect to the property that each of them contains only vertices whose states in Y are the same. Clearly, all vertices in the same subgraph must have equal states also in a predecessor of Y . Let us call each of such maximal connected subgraphs an MCS.. Let H be an MCS in which a vertex v exists whose neighbors in G all have the same state as its own. In other words, all of v's neighbors are also in H. For this vertex, clearly there is no possibility other than Y ′ (v) = Y (v). Because there is only one choice of state for v in a predecessor configuration, we call both the vertex and its containing MCS locked. We refer to all other vertices and MCSs as being unlocked (so there may exist unlocked vertices in a locked MCS). We also say that any two MCSs are neighbors whenever they contain vertices that are themselves neighbors. • No two locked MCSs are neighbors;. • Every vertex in an unlocked MCS has at least one neighbor in another unlocked MCS. 3 NP-completeness of Pre(k) for k > 1. We first note that the NP-completeness proof of Predecessor Existence for finite cellular automata [17] cannot be directly extended to k-reversible processes in graphs for k > 1. Sutner's proof only shows that there exist finite cellular automata for which Predecessor Existence is NP-complete. In other words, it depends on the vertex update rule being used in the cellular automaton. A different approach is then needed.. We present a reduction from a satisfiability problem known as 3Sat Exactly-Two. This problem is the variation of the 3Sat problem in which each clause must be satisfied by exactly two positive literals. We start by proving that 3Sat Exactly-Two is NP-complete.. Lemma 2. 3Sat Exactly-Two is NP-complete.. Proof. The problem is trivially in NP. We proceed with the reduction from another variation of the 3Sat problem, known as 3Sat Exactly-One [8] , which is NP-complete and asks whether there exists an assignment of variables satisfying each clause by exactly one positive literal.. The reduction is simple and consists of inverting all literals in all clauses of an instance S of 3Sat Exactly-One, resulting in an instance S ′ of 3Sat Exactly-Two. It is easy to check that a solution for S directly gives a solution for S ′ , and conversely, a solution for S ′ directly gives a solution for S.. Proof. Given two configurations Y and Y ′ , verifying whether Y ′ is a predecessor configuration of Y is straightforward and can be done by simulating one step of the k-reversible process starting with configuration Y ′ . Simulating one step of the process takes O(n + m) time and the final comparison between the resulting configuration and Y takes O(n) time; thus, Pre(k) is in NP. The remainder of the proof is a reduction from 3Sat Exactly-Two, which by Lemma 2 is NP-complete.. Let S be an arbitrary instance of 3Sat Exactly-Two with the M clauses c 1 , c 2 , . . . , c M and the N variables x 1 , x 2 , . . . , x N . We construct an instance (G, Y ) of Pre(k) from S as follows.. Vertex set V (G) is the union of:. • {p i,1 , . . . , p i,2k−3 }, for each variable x i in S;. • {w. • {b If x is a neighbor of u and x is a literal vertex, then we say that x is a literal neighbor of u. Similarly, if x is a neighbor of u and x is a clause vertex, then x is a clause neighbor of u.. Edge set E(G) is the union of:. We finish the construction by defining the target configuration Y : Figure 1 illustrates the case of k = 3, M = 1, and N = 3, the single clause being. For each variable in S, at most 6k−6 vertices are created, and for each clause at most 2k − 1 vertices, resulting in at most n = N (6k − 6) + M (2k − 1) vertices. Likewise, the total number of edges is at most m = N (6k − 6) + M (2k + 3). Since k is a constant, we have a polynomial-time reduction.. We proceed by showing that if S is satisfiable then Y has at least one predecessor configuration. In fact, given any satisfying truth assignment for S, we can construct a predecessor configuration Y ′ of Y in the following manner: • Y ′ (x i ) = +1, if variable x i is true in the given assignment; Figure 1 . As an example, we have let all of x 1 , ¬x 2 , and x 3 be true in the satisfying assignment.. By construction, and considering configuration Y ′ as described above, each vertex x i or ¬x i has at least k neighbors in state +1 in Y ′ and exactly k − 1 neighbors in state −1. This condition is sufficient to guarantee that every literal vertex reaches state +1 in one time step, regardless of its state in configuration Hence, given the configuration Y ′ as constructed, we see that the next configuration is precisely configuration Y . We then conclude that whenever S is satisfiable, Y has at least one predecessor configuration.. Conversely, we now show that if Y has at least one predecessor configuration then S is satisfiable.. In any predecessor configuration of Y , vertices u i,j , p i,j , w i,j , w Each vertex c ′ i has the same literal neighbors as vertex c i . As we know that some of these neighbors have state +1 in a predecessor configuration, c ′ i must have state +1, otherwise all of these neighbors would need to have state −1 in the predecessor configuration, contradicting the fact that some of them have state +1. Along with the restriction imposed by c i , we have that exactly two of the three literal vertices associated with clause c i must have state +1 in a predecessor configuration.. Hence, given any predecessor configuration of Y , we can associate with any literal the value true if its vertex has state +1, and value false otherwise. The construction guarantees that this will satisfy every clause with exactly two positive literals and that no opposite literals will have the same assignment.. We conclude that if Y has a predecessor configuration then S is satisfiable.. Corollary 4. Pre(k) on bipartite graphs is NP-complete for k > 1.. Proof. The graph constructed in the proof of Theorem 3 is bipartite for the node sets i, In this paper we have dealt with Pre(k) and #Pre(k), our denominations for the Predecessor Existence problem and its counting variation for kreversible processes. We have shown that Pre(1) is solvable in polynomial time, that Pre(k) is NP-complete for k > 1 even for bipartite graphs, and that it can be solved in polynomial time for trees. For trees we have also shown that #Pre(k) is polynomial-time solvable. We have also demonstrated the polynomial-time solvability of Pre(2) if ∆(G) ≤ 3.. We identify two problems worth investigating:. • Identify other cases in which Pre(k) can be solved in polynomial time.. • Study the complexity properties of #Pre(2) for ∆(G) ≤ 3.

O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I I O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Alterations in vasomotor control of coronary resistance vessels in remodelled myocardium of swine with a recent myocardial infarction Abstract The mechanism underlying the progressive deterioration of left ventricular (LV) dysfunction after myocardial infarction (MI) towards overt heart failure remains incompletely understood, but may involve impairments in coronary blood flow regulation within remodelled myocardium leading to intermittent myocardial ischemia. Blood flow to the remodelled myocardium is hampered as the coronary vasculature does not grow commensurate with the increase in LV mass and because extravascular compression of the coronary vasculature is increased. In addition to these factors, an increase in coronary vasomotor tone, secondary to neurohumoral activation and endothelial dysfunction, could also contribute to the impaired myocardial oxygen supply. Consequently, we explored, in a series of studies, the alterations in regulation of coronary resistance vessel tone in remodelled myocardium of swine with a 2 to 3-week-old MI. These studies indicate that myocardial oxygen balance is perturbed in remodelled myocardium, thereby forcing the myocardium to increase its oxygen extraction. These perturbations do not appear to be the result of blunted b-adrenergic or endothelial NO-mediated coronary vasodilator influences, and are opposed by an increased vasodilator influence through opening of K ATP channels. Unexpectedly, we observed that despite increased circulating levels of noradrenaline, angiotensin II and endothelin-1, a-adrenergic tone remained negligible, while the coronary vasoconstrictor influences of endogenous endothelin and angiotensin II were virtually abolished. We conclude that, early after MI, perturbations in myocardial oxygen balance are observed in remodelled myocardium. However, adaptive alterations in coronary resistance vessel control, consisting of increased vasodilator influences in conjunction with blunted vasoconstrictor influences, act to minimize the impairments of myocardial oxygen balance. Heart failure constitutes a major cardiovascular disorder of which the incidence and prevalence are increasing, principally due to an increased survival of acute myocardial infarction (MI) in conjunction with an ageing population. The mechanism underlying the progressive deterioration of left ventricular (LV) dysfunction towards overt heart failure remains incompletely understood, but may involve (1) loss of cardiomyocytes through apoptosis [75] , (2) a primary reduction in contractile function of the surviving myocardium [97] , and/or (3) alterations in extracellular matrix leading to progressive LV dilation [87] . In addition, myocardial blood flow (MBF) abnormalities, resulting in impaired myocardial O 2 delivery to the non-infarcted regions (leading to secondary contractile dysfunction and/ or enhanced apoptosis), have been suggested to contribute to the progression of LV dysfunction after MI [99] . For example, in vivo studies in rats [54, 55] and swine [108] indicate a reduction in MBF reserve of up to 35% in the surviving remodelled LV myocardium, 3-8 weeks after infarction. Furthermore, in patients with overt heart failure [94] , but also in patients with only asymptomatic LV dysfunction [95] , flow reserve is reduced in the non-stenotic myocardial regions. In line with these clinical observations, we observed in a porcine model of postinfarct remodelling that during increased O 2 -demand induced by exercise, the increase in coronary blood flow (CBF) is impaired resulting in perturbations in oxygen delivery [43, 70] . The reduction in flow reserve and the perturbed oxygen delivery during exercise are caused, at least in part, by insufficient growth of the coronary vasculature to maintain flow capacity commensurate with myocardial hypertrophy, in conjunction with a decrease in diastolic pressure time index resulting from the elevated heart rate and particularly elevated LV diastolic pressures [43] . In addition, coronary vasomotor tone may also be increased secondary to neurohumoral activation and endothelial dysfunction, further adding to the perturbations in CBF. However, little is known about the alterations in vasomotor control in coronary resistance vessels within remodelled myocardium (Fig. 1 ). For this reason we undertook a series of studies to determine whether neurohumoral (autonomic nervous system and renin-angiotensin system), local metabolic, and endothelial control mechanisms of coronary resistance vessel tone are altered in swine with remodelled myocardium produced by a recent MI.

O O O O O O O O O O O O B O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O


An analysis of the factors influencing differences in survey-reported and GPS-recorded trips Recent advances in global positioning systems (GPS) technology have resulted in a transition in household travel survey methods to test the use of GPS units to record travel details, followed by the application of an algorithm to both identify trips and impute trip purpose, typically supplemented with some level of respondent confirmation via prompted-recall surveys. As the research community evaluates this new approach to potentially replace the traditional surveyreported collection method, it is important to consider how well the GPS-recorded and algorithm-imputed details capture trip details and whether the traditional survey-reported collection method may be preferred with regards to some types of travel.This paper considers two measures of travel intensity (survey-reported and GPSrecorded) for two trip purposes (work and non-work) as dependent variables in a joint ordered response model. The empirical analysis uses a sample from the full-study of the 2009 Indianapolis regional household travel survey. Individuals in this sample provided diary details about their travel survey day as well as carried wearable GPS units for the same 24-hour period.The empirical results provide important insights regarding differences in measures of travel intensities related to the two different data collection modes (diary and GPS). The results suggest that more research is needed in the development of workplace identification algorithms, that GPS should continue to be used alongside rather than in lieu of the traditional diary approach, and that assignment of individuals to the GPS or diary survey approach should consider demographics and other characteristics. Over the past decade, the application of global positioning systems (GPS) technology within the context of household travel data collection has become commonplace. The role of GPS has primarily been for the purpose of evaluating the accuracy of trip reporting -identifying trips recorded in the GPS data streams that were not reported by the respondents in the travel diaries. Analyses of these non-reported trips have provided critical insights that have led to improvements of the survey materials and data collection methods, resulting in improved capture of trips via travel diaries.. Recent advances in GPS technology have further resulted in the introduction of lighter weight GPS units with longer battery lives, transitioning the travel survey application of GPS from in-vehicle to on-person/wearable units. This technology improvement has provided the opportunity to begin collecting GPS data for all modes of travel, particularly walking and transit. With the expansion of focus from vehicle travel to travel by all modes of travel, researchers have begun investigating in earnest the possibility of replacing paper travel diaries with GPS-recorded travel. In fact, two large-scale "proof of concept" studies are currently underway in Cincinnati and Jerusalem.. As the travel survey community evaluates the use of GPS as a primary data collection method, it is important to assess the accuracy of the GPS-recorded trips in comparison with the respondent-reported trips obtained through the traditional travel survey diary method. Efforts to date comparing survey-reported trips to GPS-recorded trips have considered the GPS data to be correct. However, GPS units themselves (and the subsequent travel extraction and activity purpose imputation algorithms) are not fail-proof -GPS traces may not capture the start of a trip if a user starts traveling before a satellite fix is obtained; the GPS signal can be lost in urban canyons or tunnels; or the GPS traces may be complete but the trip-detection algorithms may be too sensitive (over-identifying trips) or not sensitive enough (under-identifying trips).. The primary objective of this paper is to review survey-reported trips and GPS-recorded trips for the same individuals in an effort to understand how the number of trips obtained through these two methods differ. While the study does not directly shed light on which of these two sources may be closer to the "truth", or what explicitly causes the difference between surveyreported and GPS-recorded trips, it does begin to shed light on the individual, household, travel, and contextual factors that contribute to the extent of difference between the two sources. Understanding the factors underlying these differences is a necessary prelude to fine-tuning survey collection efforts in the near term and improving GPS-data collection efforts in the medium-to-long term. In this regard, the analysis begins by considering both the data sources as providing valid measures of travel intensity, albeit potentially with some measurement error. Further, both the number of survey-reported and GPS-recorded trips are modeled jointly using a copula-based framework to account for common unobserved factors that affect these two travel intensity measures. This framework enables us to determine the type as well as magnitude of dependency or the inter-relationship between these two measures of intensities of individual travel. Since the factors determining the amount of individual travel might be different for different activity purposes, two different joint models are reported for work trips of workers (a worker is defined as a person who is either employed for pay or works voluntarily) and non-work trips of all respondents. Of course, once the joint model for the two trip-making propensities is estimated, this is translated into the effects of variables on the difference of the survey-reported and GPS-recorded trips. This paper contributes to the travel survey literature in several important ways. It is the first to consider GPS data from a large-scale on-person subsample of a household travel survey (n=265 individuals). Prior papers on related topics have focused on in-vehicle GPS data and/or smaller scale wearable studies. The study also undertakes the analysis using recently collected GPS data (from 2009) that accommodates several technological advances unavailable in the GPS data collection efforts from the earlier part of this decade. Second, unlike earlier studies that have considered GPS units as providing the "true" measure of travel intensity, and have focused on trip under-reporting in surveys (see next section), the emphasis here is on the difference between survey-reported and GPS-recorded trips, without making any a priori assumptions about the accuracy of the data from the two sources. In doing so, we consider cases where the difference in trip-making between the two sources may be positive, negative, or equal (on the other hand, earlier studies have focused on the case where the difference between survey and GPS trips is negative; see, for example, Bricka and Bhat, 2006) . Finally, in this transitional time from GPS as a supporting survey method to GPS as a potential (partial or full) replacement for the travel diary survey data collection method, the current study provides insights into the factors that affect the difference between the trip capture from the two sources, which can then be used in subsequent research to further invest resources on the kinds of individuals and travel environments that result in larger magnitudes of trip differences between the two data sources. One important limitation of the current study also needs to be noted: this paper accepts the survey and GPS samples as is -considerations of self-selection bias and the representativeness of the sample are not addressed in this first, but important, effort to examine differences in trip capture between the two sources. Besides, given the focus of the current research, issues of sample representativeness are of secondary consideration.. The remainder of this paper is structured in five sections. The next section provides a summary of GPS efforts to date, while Section 3 presents an overview of the Indianapolis regional household travel survey and its descriptive GPS sample characteristics. Section 4 discusses the model structure and estimation procedure. Section 5 focuses on the empirical results. The final section summarizes the important findings from the results, and recommends specific improvements for capture of travel by both diary and GPS survey methods. This paper has examined the differences in measures of intensity of travel as obtained using survey-reported and GPS-recorded data collection methods for two trip purposes (work and nonwork) through the application of a joint ordered response model. The research models both the propensity to record trips of each purpose, as well as the propensity to have a discord between survey-reported and GPS-recorded numbers of trips. The empirical analysis in the paper uses data extracted from the 2009 Indianapolis regional household travel survey. Specifically, household members aged 16+ who both reported their travel patterns using a survey diary and who carried wearable GPS units for the same 24-hour diary period were selected for the analysis. The findings from this research can inform both survey-reported as well as GPS-recorded travel data collection approaches. In terms of work trip reporting, those respondents with a higher propensity to travel for work purposes are Caucasian, younger (under the age of 30), with lower education levels. Those employed have higher work-trip making propensities than volunteers, and within workers, those who are employed full-time have higher levels than part-time workers. Those with lower propensities to travel for work include students and individuals in households with a higher number of students, those who are employed part-time or who volunteer, and those who reported travel for a Friday. These results are consistent with the literature, validating the research approach.. Individuals characterized as Caucasian, more highly educated, and not employed for pay (retired, not employed, or volunteer) , with higher incomes living in the central business district had a higher propensity to travel for non-work purposes. On the other hand, those employed for pay had lower propensities to make non-work trips, with that propensity decreasing as the number of days worked per week increased. Again, these findings are intuitive in that those with more discretionary time would make more discretionary trips.. When considering the magnitude of differences between the survey-reported and GPSrecorded trips for both work and non-work, five main conclusions can be drawn:. 1. For the most part, the majority of trips identified for both work and non-work purposes are similar for the two data collection methods -63% of all work trips and 42% of all non-work trips -were identified to have equal numbers of trips in the survey-reported and GPS-recorded data, as indicated at the top of Tables 1 and 2 . However, where the magnitude of trip making identified through each survey method does not match, there are inconsistencies in terms of which survey method obtained more trips and which survey method obtained more accurate trip details. This suggests that improvements are needed in both methods to yield more stable trip rates for the trip purposes studied here. For this reason, we recommend that GPS-only studies be undertaken with caution until methodological improvements result in more consistent data being obtained from both sources.. 2. GPS can and should be used to obtain travel details from the younger generation -either those still in college or those who are not 1-or 2-person worker households. The results of this study show that the levels of trip reporting for work and non-work trips among this technologically savvy respondent group are the same and given their propensity to prefer technology, GPS should be deployed to this group whenever possible. GPS should also be used for those respondents characterized as heavy travelers and those more predisposed to making non-motorized trips. This includes workers who travel as part of their work, or, as indicated in this study, busy volunteers, middle-aged individuals (30-50 years of age), and those who have characteristics associated with the tendency to trip chain, as such individuals appear not to fully record their travel using the traditional survey method.. 3. Any consideration of collecting data using GPS only should be undertaken with caution, with particular care given to the specification of the trip purpose and mode detection algorithms. As indicated in this study, work trips associated with those who hold more than one job or travel to different workplaces on the travel day were not detected in the GPS data at the same levels of intensities as compared to survey data. Given that most algorithms consider the primary work locations only, respondents that work at multiple sites throughout the work day/week or work multiple-jobs at different locations report more work trips in the survey but the GPS-recorded trips reflect a portion of those as nonwork trips. Other explanations include the possibility that the buffers used to compare the work location to the GPS-recorded trip location are too tight (i.e., they need to be expanded to capture off-site parking) or that the work locations are within urban canyons or other locations where satellite receptivity is poor and those GPS data points are excluded from analysis. In addition, travel by transit and non-motorized modes were also associated with significant differences in trips reported, indicating that the trip detection algorithms require review.. 4. The results also suggest that the traditional survey approach may be better suited for the elderly, retirees, and those who can be characterized as having more leisurely travel.. More research is needed to ascertain why these individuals report more survey trips than GPS, with a concern that perhaps these respondents are not carrying the GPS units for all trips, or that they are reporting in their surveys "typical" travel as compared to actual travel on the assigned travel day. This group also includes those new to the region, and who perhaps travel more tentatively than those who have lived in the region for more than two years. The tentative driving style of the newcomers may result in more GPSrecorded non-work trips as compared to survey-reported data due to over-detection of stops in the GPS data streams. Given the cost of deploying GPS, combined with the fact that the traditional survey method appears to capture trips sufficiently for these individuals, this may be a group to target for the traditional and less-costly diary/survey approach.. 5. An interesting finding of this research, and one that requires further study, is the phenomenon of Friday travel. As reported herein, individuals who travel on Fridays tend to travel more for non-work purposes than work purposes. The work trips obtained by both the survey and GPS are consistent with each other and the literature, which indicates that both data collection methods tend to match at lower trip making intensities. For nonwork travel, survey-reported trips tend to be fewer than GPS-recorded trips, suggesting that there is more trip chaining and special travel patterns on Fridays as compared to other weekdays. For this reason, we recommend that when GPS is used in a travel survey, a higher proportion of Friday travelers be equipped with the GPS units to better capture and measure travel patterns on this specific day of the week.. In summary, this paper has examined the propensities and magnitude of trip making as reported through the survey data and as recorded using GPS units for both work and non-work purposes. The findings suggest that instead of an either/or approach to data collection, that the two survey methods complement each other and should be used in tandem. Any study considering GPS-collection only should strongly consider the details used to develop the algorithms to assign trip purpose, as our results show that the use of primary workplace location only tends to under-record work trips. However, GPS should be considered as the data collection method when dealing with the younger, more technology savvy individuals as well as those that have high travel propensities or characteristics associated with trip chaining, in order to ensure that all trip details are recorded. However, for the elderly and more leisurely travelers, the traditional survey method is recommended

O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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Quasi-Newton methods for implicit black-box FSI coupling In this paper we introduce a new multi-vector update quasi-Newton (MVQN) method for implicit coupling of partitioned, transient FSI solvers. The new quasi-Newton method facilitates the use of 'black-box' eld solvers and under certain circumstances can be demonstrated to provide Newton-like convergence behaviour for strongly coupled FSI benchmark problems. We demonstrate the superior convergence behaviour and robust nature of the MVQN method compared to other well known quasi-Newton coupling schemes, including the least squares reduced order modelling (IBQN-LS) scheme, the classical rank-1 update Broyden's method and xed point iterations with dynamic relaxation.The quasi-Newton methods are analysed on a suite of strongly coupled FSI problems, including but not limited to, internal, incompressible ow through a exible tube where the solid density is an order of magnitude lower than the uid density. With an ever increasing availability and size of computational resources at the disposal of researchers, multiphysics simulations are becoming an ever present reality. Fluid-structure interactions (FSI), a subclass of multiphysics problems, have in particular received much attention over the past several years. The main reason for this is the wide range of problems within the elds of engineering and life sciences that involve FSI. These problems, to mention but a few, range from utter prediction in aeroelasticity [15, 25] , parachute dynamics [26, 28] and blood ow through the vascular system [29, 32, 33] .. There are two main approaches to solving the FSI problem, namely the monolithic approach or the partitioned approach. The monolithic approach is to solve all equations, including those related to the interface and computational domains in a single unied solver, typically involving some variant of Newton's method [4, 21] . Alternatively, a partitioned solver utilises separate eld solvers for the uid and solid domains, with these separate domains then coupled along the interface. Partitioned schemes can then be solved explicitly or allow for sub-iterations or implicit coupling in order to guarantee satisfaction of the governing equations along the interface. has been coined as the added-mass eect [7, 17] . The general trend in literature, in order to obtain a stable and robust solution procedure for these class of problems, has been that monolithic solvers are required [9, 20, 23] . While there are merits to this argument, it does however remain dicult to justify the enormous initial investment required in developing a monolithic solver. This is especially relevant considering the large availability of optimised uid and solid solvers, which the use of monolithic solution procedures completely preclude. Equally, segregated solvers allow for the solution of smaller systems as opposed to one large unied set of equations, which unless properly preconditioned can be very expensive [20] . Partitioned solvers further allow for each of the sub-domain elds to be solved using eld specic discretisation and solution schemes.. Several useful contributions have been made with regards to partitioned incompressible FSI technology, ranging from articial compressibility [22, 11] , robin boundary conditions [2] , and computing the exact Jacobians or sensitivities [13, 16] of the cross-coupling terms between the uid and solid domains. While these contributions represent distinct steps in allowing for the re-use of existing solvers, they are limited in the sense that they require access to the source of one or both of the domain solvers.. In this paper we are interested in the application of FSI using 'black-box' solvers, where no access to source code is available. FSI solvers based on black-box solvers mean that most uid or solid solvers, including commercially available solvers can be used. The general understanding however is that black-box coupling algorithms have only limited applicability to the range of complexity of problems to which they can be applied, and are often insucient when dealing with FSI problems where the solid to uid mass ratios are close to unity.. Several coupling algorithms have been proposed which facilitate the use of black-box eld solvers. They typically involve xed point iterations, with augmentations including dynamic relaxation or other similar acceleration techniques [23] . For strongly coupled problems the convergence behaviour is often poor, if convergence is obtained at all. Quasi-Newton methods, whereby the coupling Jacobians are approximated rather than explicitly computed have shown promising results. Most notably of these is the use of nite-dierencing (FD) to approximate the matrixvector product for Newton-Krylov solution methods [24] .. While Newton-like convergence behaviour has been observed, the FD methods have been demonstrated to be sensitive to the choice of step size. They also require several solid and uid solver calls within a given sub-iteration, often making them too expensive, especially when compared to monolithic solution schemes [23] .. In this paper we will introduce a new quasi-Newton method designed for the coupling of transient, black-box FSI solvers. The coupling method is based on a multivector iterative updating scheme requiring only observations of the interface tractions and displacements. It allows for the ecient and robust solution of strongly coupled transient FSI problems. The new proposed coupling scheme is heuristic independent and can be applied to a wide range of complex FSI problems, which can be shown under certain circumstances to provide Newton-like convergence behaviour.. We will compare the new proposed scheme to the reduced order modelling (ROM) scheme introduced by Vierendeels et al. [30] , one of the more promising black-box coupling schemes. We will illustrate how the ROM coupling scheme can be viewed as a minimum norm solution scheme.. This provides insights into why the method works comparatively well and provides some explanations as to its behaviour with regards to the number of retained observations in time. We will further compare these methods to the classical Broyden's quasi-Newton method and the popular (though somewhat limited) Aitken's dynamic relaxation.. The outline of the remainder of the paper is as follows.. In Section 2 we describe the general black-box FSI problem within a block-Newton framework. Section 4 introduces the multi-vector Jacobian update scheme and briey outlines the other quasi-Newton methods investigated in this paper. Lastly, the performance of the quasi-Newton methods is analysed in Section 5 using four popular benchmark problems covering a wide range of complexity, including incompressible ow through a exible tube where the solid density is an order of magnitude lower than the uid density.

B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Construction of universal one-way hash functions: Tree hashing revisited We present a binary tree based parallel algorithm for extending the domain of a universal one-way hash function (UOWHF). For t 2, our algorithm extends the domain from the set of all n-bit strings to the set of all ((2 t − 1)(n − m) + m)-bit strings, where m is the length of the message digest. The associated increase in key length is 2m bits for t = 2; m(t + 1) bits for 3 t 6 and m × (t + log 2 (t − 1) ) bits for t 7. Let F = {h k } k∈K be a family of functions where each h k : X → Y . The set K is said to be the set of keys and F is said to be a keyed family of functions. Consider the following adversarial game with respect to F: the adversary outputs an x ∈ X; is then given a randomly chosen key k ∈ K; and has to output an x ∈ X such that x = x and h k (x) = h k (x ) . The family F is said to be a universal one-way hash function (UOWHF) if it is infeasible for an adversary to win this game. The concept of UOWHF was introduced by Naor and Yung [6] and used by them to show that it is possible to construct secure digital signature schemes based on 1-1, one-way functions.. The notion of collision resistant hash function (CRHF) is more common in cryptography. In this notion, the adversary is first given a k and then has to output x and x such that x = x and h k (x) = h k (x ) . Note that in the case of UOWHF, the adversary has to commit to the input x even before he knows the function for which a collision is to be found. Intuitively, this makes the adversary's task more difficult and hence UOWHF is considered to be a weaker primitive. In fact, Simon [9] has shown that there is an oracle relative to which UOWHFs exist but not CRHFs.. A systematic study of construction methods for UOWHF was undertaken by Bellare and Rogaway [1] . The approach taken in [1] is to start with a UOWHF {h k } k∈K , where the domain X 1 of h k consists of short fixed length strings and then obtain methods to construct another UOWHF {H p } p∈P where the domain X 2 of H p consists of long fixed length strings. Such a method is called a domain extending algorithm. Further, it is shown in [1] , that extending the domain also requires an increase in the length of the keys.. The most important method in [1] is a tree based construction. For binary trees with t 2 levels, the BR construction extends the domain from n-bit strings to ((2 t − 1)(n − m) + m)-bit strings and requires an associated increase in key length by 2m(t − 1) bits, where the range Y of the hash functions consists of m-bit strings. Later, Shoup [8] provided a sequential construction which requires an increase in key length of mt bits for extending the domain by the same amount. Thus, the sequential algorithm has a smaller key length expansion compared to the earlier binary tree based algorithm. Mironov [5] provided an alternative proof of correctness of Shoup's result and showed that the key length expansion is the best possible for any sequential algorithm.. The essential computational task in a domain extending algorithm consists of executing h k several times. In a binary tree based algorithm, the different invocations of h k are organised in a binary tree. This opens up the possibility of parallel execution using more than one processors. Such a parallel algorithm will have several parallel rounds. In each round, all (or some) of the processors would operate in parallel and produce outputs to be used in the next round. Since more than one processors work simultaneously, the number of parallel rounds will be less than the total number of invocations of h k made by the algorithm. On the other hand, a sequential algorithm using one processor will require time proportional to the total number of invocations of h k . There are standard ways to convert a binary tree structure into an efficient parallel algorithm (see [3] ). Thus, even though the binary tree algorithm of [1] has a larger key length expansion compared to the sequential algorithm of [8] , it will be more efficient to implement when more than one processors are available.. In this work, we show that a modification of the binary tree algorithm of [1] results in a smaller key length expansion while retaining the advantage of parallelism. The key length expansion of our algorithm is 2m bits for t = 2; m(t + 1) bits for 3 t 6 and m × (t + log 2 (t − 1) ) bits for t 7. This is an improvement over the algorithm in [1] though it is more than the key length expansion achieved in [8] . The improvement in key length expansion over [1] is achieved by combining the construction in [1] with the construction in [8] . The proof of correctness of our algorithm employs the technique used in [5] rather than the technique in [8] . In this paper, we have considered the problem of extending the domain of a UOWHF using a binary tree algorithm. As shown in [1] this requires an expansion in the length of the key to the hash function. To extend the domain from n-bit strings to ((2 t − 1)(n − m) + m)-bit strings, our algorithm makes a key length expansion of 2m bits for t = 2; m(t + 1) bits for 3 t 6 and m(t + log 2 (t − 1) ) for t 7 where m is the length of the message digest. The binary tree algorithm of Bellare and Rogaway [1] requires a key length expansion of 2m(t − 1) with the same parameters. Hence, the key length expansion in our algorithm is smaller. However, it is greater than the sequential algorithm due to Shoup [8] , which requires a key length expansion of mt for the same parameters. On the other hand, the advantage of the BR and our binary tree algorithm is that it is parallelizable while Shoup's algorithm is not.

O O O O O O O O O O O O B I O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O