Problem:
Score : 600 points

Problem Statement

10^9 contestants, numbered 1 to 10^9, will compete in a competition.
There will be two contests in this competition.

The organizer prepared N problems, numbered 1 to N, to use in these contests.
When Problem i is presented in a contest, it will be solved by all contestants from Contestant L_i to Contestant R_i (inclusive), and will not be solved by any other contestants.

The organizer will use these N problems in the two contests.
Each problem must be used in exactly one of the contests, and each contest must have at least one problem.

The joyfulness of each contest is the number of contestants who will solve all the problems in the contest.
Find the maximum possible total joyfulness of the two contests.

Constraints

2 \leq N \leq 10^5

1 \leq L_i \leq R_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
L_1 R_1
L_2 R_2
\vdots
L_N R_N

Output

Print the maximum possible total joyfulness of the two contests.

Sample Input 1

4
4 7
1 4
5 8
2 5

Sample Output 1

6

The optimal choice is:

Use Problem 1 and 3 in the first contest. Contestant 5, 6, and 7 will solve both of them, so the joyfulness of this contest is 3.

Use Problem 2 and 4 in the second contest. Contestant 2, 3, and 4 will solve both of them, so the joyfulness of this contest is 3.

The total joyfulness of these two contests is 6. We cannot make the total joyfulness greater than 6.

Sample Input 2

4
1 20
2 19
3 18
4 17

Sample Output 2

34

Sample Input 3

10
457835016 996058008
456475528 529149798
455108441 512701454
455817105 523506955
457368248 814532746
455073228 459494089
456651538 774276744
457667152 974637457
457293701 800549465
456580262 636471526

Sample Output 3

540049931

Solution in Rust (1.15.1) (judge verdict: Accepted)

pub trait Readable {
    type Output;
    fn words_count() -> usize;
    fn read_words(words: &[&str]) -> Result<Self::Output, String>;
}
#[macro_export]
macro_rules! readable {
    ( $ t : ty , $ words_count : expr , |$ words : ident | $ read_words : expr ) => {
        impl Readable for $t {
            type Output = $t;
            fn words_count() -> usize {
                $words_count
            }
            fn read_words($words: &[&str]) -> Result<$t, String> {
                Ok($read_words)
            }
        }
    };
}
readable!((), 1, |_ss| ());
readable!(String, 1, |ss| ss[0].to_string());
impl Readable for char {
    type Output = char;
    fn words_count() -> usize {
        1
    }
    fn read_words(words: &[&str]) -> Result<char, String> {
        let chars: Vec<char> = words[0].chars().collect();
        if chars.len() == 1 {
            Ok(chars[0])
        } else {
            Err(format!("cannot parse `{}` as a char", words[0]))
        }
    }
}
pub struct Chars();
impl Readable for Chars {
    type Output = Vec<char>;
    fn words_count() -> usize {
        1
    }
    fn read_words(words: &[&str]) -> Result<Vec<char>, String> {
        Ok(words[0].chars().collect())
    }
}
macro_rules ! impl_readable_for_ints { ( $ ( $ t : ty ) * ) => { $ ( impl Readable for $ t { type Output = Self ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result <$ t , String > { use std :: str :: FromStr ; <$ t >:: from_str ( words [ 0 ] ) . map_err ( | _ | { format ! ( "cannot parse `{}` as {}" , words [ 0 ] , stringify ! ( $ t ) ) } ) } } ) * } ; }
impl_readable_for_ints ! ( i8 u8 i16 u16 i32 u32 i64 u64 isize usize f32 f64 ) ;
macro_rules ! define_one_origin_int_types { ( $ new_t : ident $ int_t : ty ) => { # [ doc = " Converts 1-origin integer into 0-origin when read from stdin." ] # [ doc = "" ] # [ doc = " # Example" ] # [ doc = "" ] # [ doc = " ```no_run" ] # [ doc = " # #[macro_use] extern crate atcoder_snippets;" ] # [ doc = " # use atcoder_snippets::read::*;" ] # [ doc = " // Stdin: \"1\"" ] # [ doc = " read!(a = usize_);" ] # [ doc = " assert_eq!(a, 0);" ] # [ doc = " ```" ] # [ allow ( non_camel_case_types ) ] pub struct $ new_t ; impl Readable for $ new_t { type Output = $ int_t ; fn words_count ( ) -> usize { 1 } fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { <$ int_t >:: read_words ( words ) . map ( | n | n - 1 ) } } } ; ( $ new_t : ident $ int_t : ty ; $ ( $ inner_new_t : ident $ inner_int_t : ty ) ;* ) => { define_one_origin_int_types ! ( $ new_t $ int_t ) ; define_one_origin_int_types ! ( $ ( $ inner_new_t $ inner_int_t ) ;* ) ; } ; }
define_one_origin_int_types ! ( u8_ u8 ; u16_ u16 ; u32_ u32 ; u64_ u64 ; usize_ usize ) ;
macro_rules ! impl_readable_for_tuples { ( $ t : ident $ var : ident ) => ( ) ; ( $ t : ident $ var : ident ; $ ( $ inner_t : ident $ inner_var : ident ) ;* ) => { impl_readable_for_tuples ! ( $ ( $ inner_t $ inner_var ) ;* ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,*> Readable for ( $ t , $ ( $ inner_t ) ,* ) { type Output = ( <$ t >:: Output , $ ( <$ inner_t >:: Output ) ,* ) ; fn words_count ( ) -> usize { let mut n = <$ t >:: words_count ( ) ; $ ( n += <$ inner_t >:: words_count ( ) ; ) * n } # [ allow ( unused_assignments ) ] fn read_words ( words : & [ & str ] ) -> Result < Self :: Output , String > { let mut start = 0 ; let $ var = <$ t >:: read_words ( & words [ start .. start +<$ t >:: words_count ( ) ] ) ?; start += <$ t >:: words_count ( ) ; $ ( let $ inner_var = <$ inner_t >:: read_words ( & words [ start .. start +<$ inner_t >:: words_count ( ) ] ) ?; start += <$ inner_t >:: words_count ( ) ; ) * Ok ( ( $ var , $ ( $ inner_var ) ,* ) ) } } } ; }
impl_readable_for_tuples ! ( T8 x8 ; T7 x7 ; T6 x6 ; T5 x5 ; T4 x4 ; T3 x3 ; T2 x2 ; T1 x1 ) ;
pub trait ReadableFromLine {
    type Output;
    fn read_line(line: &str) -> Result<Self::Output, String>;
}
fn split_into_words(line: &str) -> Vec<&str> {
    #[allow(deprecated)]
    line.trim_right_matches('\n').split_whitespace().collect()
}
impl<T: Readable> ReadableFromLine for T {
    type Output = T::Output;
    fn read_line(line: &str) -> Result<T::Output, String> {
        let words = split_into_words(line);
        if words.len() != T::words_count() {
            return Err(format!(
                "line `{}` has {} words, expected {}",
                line,
                words.len(),
                T::words_count()
            ));
        }
        T::read_words(&words)
    }
}
macro_rules ! impl_readable_from_line_for_tuples_with_from_iterator { ( $ u : ident : $ ( + $ bound : path ) * => $ seq_in : ty , $ seq_out : ty ; $ t : ident $ var : ident ) => { impl <$ u : Readable > ReadableFromLine for $ seq_in where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = $ seq_out ; fn read_line ( line : & str ) -> Result <$ seq_out , String > { let n = $ u :: words_count ( ) ; let words = split_into_words ( line ) ; if words . len ( ) % n != 0 { return Err ( format ! ( "line `{}` has {} words, expected multiple of {}" , line , words . len ( ) , n ) ) ; } let mut result = Vec :: new ( ) ; for chunk in words . chunks ( n ) { match $ u :: read_words ( chunk ) { Ok ( v ) => result . push ( v ) , Err ( msg ) => { let flagment_msg = if n == 1 { format ! ( "word {}" , result . len ( ) ) } else { let l = result . len ( ) ; format ! ( "words {}-{}" , n * l + 1 , ( n + 1 ) * l ) } ; return Err ( format ! ( "{} of line `{}`: {}" , flagment_msg , line , msg ) ) ; } } } Ok ( result . into_iter ( ) . collect ( ) ) } } impl < T : Readable , $ u : Readable > ReadableFromLine for ( T , $ seq_in ) where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = ( T :: Output , $ seq_out ) ; fn read_line ( line : & str ) -> Result < Self :: Output , String > { let n = T :: words_count ( ) ; # [ allow ( deprecated ) ] let trimmed = line . trim_right_matches ( '\n' ) ; let words_and_rest : Vec <& str > = trimmed . splitn ( n + 1 , ' ' ) . collect ( ) ; if words_and_rest . len ( ) < n { return Err ( format ! ( "line `{}` has {} words, expected at least {}" , line , words_and_rest . len ( ) , n ) ) ; } let words = & words_and_rest [ .. n ] ; let empty_str = "" ; let rest = words_and_rest . get ( n ) . unwrap_or ( & empty_str ) ; Ok ( ( T :: read_words ( words ) ?, <$ seq_in >:: read_line ( rest ) ? ) ) } } } ; ( $ u : ident : $ ( + $ bound : path ) * => $ seq_in : ty , $ seq_out : ty ; $ t : ident $ var : ident , $ ( $ inner_t : ident $ inner_var : ident ) ,+ ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : $ ( + $ bound ) * => $ seq_in , $ seq_out ; $ ( $ inner_t $ inner_var ) ,+ ) ; impl <$ t : Readable , $ ( $ inner_t : Readable ) ,+ , $ u : Readable > ReadableFromLine for ( $ t , $ ( $ inner_t ) ,+ , $ seq_in ) where <$ u as Readable >:: Output : Sized $ ( + $ bound ) * { type Output = ( $ t :: Output , $ ( $ inner_t :: Output ) ,+ , $ seq_out ) ; fn read_line ( line : & str ) -> Result < Self :: Output , String > { let mut n = $ t :: words_count ( ) ; $ ( n += $ inner_t :: words_count ( ) ; ) + # [ allow ( deprecated ) ] let trimmed = line . trim_right_matches ( '\n' ) ; let words_and_rest : Vec <& str > = trimmed . splitn ( n + 1 , ' ' ) . collect ( ) ; if words_and_rest . len ( ) < n { return Err ( format ! ( "line `{}` has {} words, expected at least {}" , line , words_and_rest . len ( ) , n ) ) ; } let words = & words_and_rest [ .. n ] ; let empty_str = "" ; let rest = words_and_rest . get ( n ) . unwrap_or ( & empty_str ) ; let ( $ var , $ ( $ inner_var ) ,* ) = < ( $ t , $ ( $ inner_t ) ,+ ) >:: read_words ( words ) ?; Ok ( ( $ var , $ ( $ inner_var ) ,* , <$ seq_in >:: read_line ( rest ) ? ) ) } } } ; }
#[macro_export]
macro_rules ! readable_collection { ( $ u : ident => $ collection_in : ty , $ collection_out : ty ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : => $ collection_in , $ collection_out ; T8 x8 , T7 x7 , T6 x6 , T5 x5 , T4 x4 , T3 x3 , T2 x2 , T1 x1 ) ; } ; ( $ u : ident : $ ( $ bound : path ) ,* => $ collection_in : ty , $ collection_out : ty ) => { impl_readable_from_line_for_tuples_with_from_iterator ! ( $ u : $ ( + $ bound ) * => $ collection_in , $ collection_out ; T8 x8 , T7 x7 , T6 x6 , T5 x5 , T4 x4 , T3 x3 , T2 x2 , T1 x1 ) ; } }
readable_collection ! ( U => Vec < U >, Vec < U :: Output > ) ;
readable_collection ! ( U => std :: collections :: VecDeque < U >, std :: collections :: VecDeque < U :: Output > ) ;
readable_collection ! ( U : Eq , std :: hash :: Hash => std :: collections :: HashSet < U >, std :: collections :: HashSet < U :: Output > ) ;
readable_collection ! ( U : Ord => std :: collections :: BTreeSet < U >, std :: collections :: BTreeSet < U :: Output > ) ;
readable_collection ! ( U : Ord => std :: collections :: BinaryHeap < U >, std :: collections :: BinaryHeap < U :: Output > ) ;
pub fn read<T: ReadableFromLine>() -> T::Output {
    let mut line = String::new();
    std::io::stdin().read_line(&mut line).unwrap();
    T::read_line(&line).unwrap()
}
#[macro_export]
macro_rules ! read { ( ) => { let mut line = String :: new ( ) ; std :: io :: stdin ( ) . read_line ( & mut line ) . unwrap ( ) ; } ; ( $ pat : pat = $ t : ty ) => { let $ pat = read ::<$ t > ( ) ; } ; ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { read ! ( ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) ) ; } ; }
#[macro_export]
macro_rules ! readls { ( $ ( $ pat : pat = $ t : ty ) ,+ ) => { $ ( read ! ( $ pat = $ t ) ; ) * } ; }
pub fn readx<T: ReadableFromLine>() -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result = stdin
        .lock()
        .lines()
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    result
}
#[macro_export]
macro_rules ! readx_loop { ( |$ pat : pat = $ t : ty | $ body : expr ) => { { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; for line in stdin . lock ( ) . lines ( ) { let line = line . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } } ; ( |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readx_loop ! ( | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub fn readn<T: ReadableFromLine>(n: usize) -> Vec<T::Output> {
    use std::io::{self, BufRead};
    let stdin = io::stdin();
    let result: Vec<T::Output> = stdin
        .lock()
        .lines()
        .take(n)
        .map(|line_result| {
            let line = line_result.expect("read from stdin failed");
            T::read_line(&line).unwrap()
        })
        .collect();
    if result.len() < n {
        panic!(
            "expected reading {} lines, but only {} lines are read",
            n,
            result.len()
        );
    }
    result
}
#[macro_export]
macro_rules ! readn_loop { ( $ n : expr , |$ pat : pat = $ t : ty | $ body : expr ) => { { use std :: io :: BufRead ; let stdin = std :: io :: stdin ( ) ; let mut lock = stdin . lock ( ) ; for _ in 0 ..$ n { let mut line = String :: new ( ) ; lock . read_line ( & mut line ) . expect ( "read from stdin failed" ) ; let $ pat = <$ t >:: read_line ( & line ) . unwrap ( ) ; $ body } } } ; ( $ n : expr , |$ ( $ pat : pat = $ t : ty ) ,*| $ body : expr ) => { readn_loop ! ( $ n , | ( $ ( $ pat ) ,* ) = ( $ ( $ t ) ,* ) | $ body ) ; } ; }
pub trait Words {
    fn read<T: Readable>(&self) -> T::Output;
}
impl<'a> Words for [&'a str] {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(self).unwrap()
    }
}
impl<'a> Words for &'a str {
    fn read<T: Readable>(&self) -> T::Output {
        T::read_words(&[self]).unwrap()
    }
}

#[macro_export]
#[cfg(local)]
macro_rules! dbg {
    ( ) => {{
        use std::io::{self, Write};
        writeln!(io::stderr(), "{}: dbg", line!()).unwrap();
    }};
    ( $ e : expr ) => {{
        use std::io::{self, Write};
        let result = $e;
        writeln!(
            io::stderr(),
            "{}: {} = {:?}",
            line!(),
            stringify!($e),
            result
        )
            .unwrap();
        result
    }};
}
#[macro_export]
#[cfg(not(local))]
macro_rules! dbg {
    ( ) => {};
    ( $ e : expr ) => {
        $e
    };
}

pub struct SegmentTree<T: Clone, F: Fn(&T, &T) -> T> {
    heap: Vec<T>,
    identity: T,
    aggregate: F,
}
mod segment_tree_internal {
    pub fn parent_of(heap_index: usize) -> usize {
        (heap_index - 1) / 2
    }
    pub fn left_of(heap_index: usize) -> usize {
        heap_index * 2 + 1
    }
    pub fn right_of(heap_index: usize) -> usize {
        left_of(heap_index) + 1
    }
    pub fn is_root_or_right(heap_index: usize) -> bool {
        heap_index % 2 == 0
    }
}
impl<T: Clone, F: Fn(&T, &T) -> T> SegmentTree<T, F> {
    pub fn new(len: usize, identity: T, aggregate: F) -> SegmentTree<T, F> {
        SegmentTree {
            heap: vec![identity.clone(); (len * 2).saturating_sub(1)],
            identity: identity,
            aggregate: aggregate,
        }
    }
    pub fn from_vec(mut items: Vec<T>, identity: T, aggregate: F) -> SegmentTree<T, F> {
        use self::segment_tree_internal::*;
        let node_count = items.len().saturating_sub(1);
        let mut heap = Vec::with_capacity(node_count + items.len());
        unsafe {
            heap.set_len(node_count);
        }
        heap.append(&mut items);
        for i in (0..node_count).rev() {
            heap[i] = aggregate(&heap[left_of(i)], &heap[right_of(i)]);
        }
        SegmentTree {
            heap: heap,
            identity: identity,
            aggregate: aggregate,
        }
    }
    fn node_count(&self) -> usize {
        self.heap.len() / 2
    }
    pub fn len(&self) -> usize {
        self.heap.len() - self.node_count()
    }
    pub fn is_empty(&self) -> bool {
        self.len() == 0
    }
    pub fn items(&self) -> &[T] {
        &self.heap[self.node_count()..self.node_count() + self.len()]
    }
    pub fn query(&self, range: std::ops::Range<usize>) -> Option<T> {
        if self.len() <= range.start {
            return None;
        }
        if self.len() < range.end {
            return None;
        }
        if range.start <= range.end {
            Some(self.aggregate_interval(
                self.node_count() + range.start,
                self.node_count() + range.end,
                self.identity.clone(),
                self.identity.clone(),
            ))
        } else {
            None
        }
    }
    fn aggregate_interval(
        &self,
        mut heap_start: usize,
        mut heap_end: usize,
        mut acc_left: T,
        mut acc_right: T,
    ) -> T {
        use self::segment_tree_internal::*;
        if heap_end == heap_start {
            (self.aggregate)(&acc_left, &acc_right)
        } else {
            if is_root_or_right(heap_start) {
                acc_left = (self.aggregate)(&acc_left, &self.heap[heap_start]);
                heap_start += 1;
            }
            if is_root_or_right(heap_end) {
                heap_end -= 1;
                acc_right = (self.aggregate)(&self.heap[heap_end], &acc_right);
            }
            self.aggregate_interval(
                parent_of(heap_start),
                parent_of(heap_end),
                acc_left,
                acc_right,
            )
        }
    }
    pub fn get_mut(&mut self, index: usize) -> Option<SegmentTreeItemRef<T, F>> {
        if index < self.len() {
            let heap_index = self.node_count() + index;
            Some(SegmentTreeItemRef {
                tree: self,
                heap_index: heap_index,
            })
        } else {
            None
        }
    }
    pub fn at(&mut self, index: usize) -> SegmentTreeItemRef<T, F> {
        self.get_mut(index).unwrap()
    }
    fn update_ancestors(&mut self, heap_index: usize) {
        use self::segment_tree_internal::*;
        if heap_index != 0 {
            let parent = parent_of(heap_index);
            let l = left_of(parent);
            let r = right_of(parent);
            self.heap[parent] = (self.aggregate)(&self.heap[l], &self.heap[r]);
            self.update_ancestors(parent);
        }
    }
}
pub struct SegmentTreeItemRef<'a, T: 'a + Clone, F: 'a + Fn(&T, &T) -> T> {
    tree: &'a mut SegmentTree<T, F>,
    heap_index: usize,
}
impl<'a, T, F> std::ops::Deref for SegmentTreeItemRef<'a, T, F>
where
    T: Clone,
    F: Fn(&T, &T) -> T,
{
    type Target = T;
    fn deref(&self) -> &T {
        &self.tree.heap[self.heap_index]
    }
}
impl<'a, T, F> std::ops::DerefMut for SegmentTreeItemRef<'a, T, F>
where
    T: Clone,
    F: Fn(&T, &T) -> T,
{
    fn deref_mut(&mut self) -> &mut T {
        &mut self.tree.heap[self.heap_index]
    }
}
impl<'a, T, F> Drop for SegmentTreeItemRef<'a, T, F>
where
    T: Clone,
    F: Fn(&T, &T) -> T,
{
    fn drop(&mut self) {
        self.tree.update_ancestors(self.heap_index);
    }
}
pub trait IteratorExtForSegmentTree: Sized + Iterator
where
    Self::Item: Clone,
{
    fn segment_tree<F>(self, identity: Self::Item, aggregate: F) -> SegmentTree<Self::Item, F>
    where
        F: Fn(&Self::Item, &Self::Item) -> Self::Item,
    {
        SegmentTree::from_vec(self.collect(), identity, aggregate)
    }
}
impl<I: Iterator> IteratorExtForSegmentTree for I where <I as Iterator>::Item: Clone {}

use std::cmp;

fn main() {
    read!(n = usize);
    let mut itvs = readx::<(u32_, u32)>();
    itvs.sort_by_key(|&(l, _)| l);
    let rs: Vec<u32> = itvs.iter().map(|&(_, r)| r).collect();
    let rmq = SegmentTree::from_vec(rs, u32::max_value(), |&a, &b| cmp::min(a, b));
    let ans1 = (1..n).map(|end| {
        let l1 = itvs[end-1].0;
        let r1 = rmq.query(0..end).unwrap();
        let l2 = itvs[n-1].0;
        let r2 = rmq.query(end..n).unwrap();
        r1.saturating_sub(l1) + r2.saturating_sub(l2)
    }).max().unwrap();

    let ans2 = (0..n).map(|i| {
        let n1 = itvs[i].1 - itvs[i].0;
        let l = if i == n-1 { itvs[n-2].0 } else { itvs[n-1].0 };
        let r = if i == n-1 {
            rmq.query(0..n).unwrap()
        } else {
            cmp::min(rmq.query(0..i).unwrap(), rmq.query(i+1..n).unwrap())
        };
        n1 + r.saturating_sub(l)
    }).max().unwrap();
    println!("{}", cmp::max(ans1, ans2));
}