We revisit the Subset Sum problem over the finite cyclic group Z_m for some given integer m. A series of recent works has provided asymptotically optimal algorithms for this problem under the Strong Exponential Time Hypothesis. Koiliaris and Xu (SODA'17, TALG'19) gave a deterministic algorithm running in time O(m^{5/4}), which was later improved to O(m \log^7 m) randomized time by Axiotis et al. (SODA'19). In this work, we present two simple algorithms for the Modular Subset Sum problem running in near-linear time in m, both efficiently implementing Bellman's iteration over Z_m. The first one is a randomized algorithm running in time O(m \log^2 m), that is based solely on rolling hash and an elementary data-structure for prefix sums; to illustrate its simplicity we provide a short and efficient implementation of the algorithm in Python. Our second solution is a deterministic algorithm running in time O(m\ polylog\ m), that uses dynamic data structures for string manipulation. We further show that the techniques developed in this work can also lead to simple algorithms for the All Pairs Non-Decreasing Paths Problem (APNP) on undirected graphs, matching the asymptotically optimal running time of O(n^2) provided in the recent work of Duan et al. (ICALP'19).
We revisit the Subset Sum problem over the finite cyclic group Z_m for some given integer m. A series of recent works has provided near-optimal algorithms for this problem under the Strong Exponential Time Hypothesis. Koiliaris and Xu (SODA'17, TALG'19) gave a deterministic algorithm running in time O(m^{5/4}), which was later improved to O(m \log^7 m) randomized time by Axiotis et al. (SODA'19). In this work, we present two simple algorithms for the Modular Subset Sum problem running in near-linear time in m, both efficiently implementing Bellman's iteration over Z_m. The first one is a randomized algorithm running in time O(m \log^2 m), that is based solely on rolling hash and an elementary data-structure for prefix sums; to illustrate its simplicity we provide a short and efficient implementation of the algorithm in Python. Our second solution is a deterministic algorithm running in time O(m\ polylog\ m), that uses dynamic data structures for string manipulation. We further show that the techniques developed in this work can also lead to simple algorithms for the All Pairs Non-Decreasing Paths Problem (APNP) on undirected graphs, matching the near-optimal running time of O(n^2) provided in the recent work of Duan et al. (ICALP'19).
R
asymptotically optimal
near-optimal
R
asymptotically optimal
near-optimal