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Finding a Shortest $M$-link Path in a Monge Directed Acyclic Graph

Joy Z. Wan

TL;DR

This work addresses finding a shortest $M$-link path in a Monge DAG with submodular edge lengths, a problem with broad algorithmic implications. The authors introduce a contract-and-conquer framework that blends a parsimonious dynamic programming, parametric shortest paths, and iterative contractions to achieve $O\left( \sqrt{NM\left( N-M\right) \log\left( N-M\right)} \right)$ time and $O\left( N \right)$ space, the first sub-$NM$ time algorithm with linear space. Their analysis covers both polylogarithmic and near-extremal $M$ regimes, delivering $O\left( N\cdot \text{poly}(\log N) \right)$ performance in the $O(\log N)$ and $N-O(\log N)$ cases and offering a clear comparison to prior strongly polynomial methods. The approach is conceptually simple, relies on elementary decreasing-and-conquer ideas, and opens avenues for hybrid strategies that adapt to the relative size of $M$ and $N-M$ in practice.

Abstract

A Monge directed acyclic graph (DAG) $G$ on the nodes $1,2,\cdots,N$ has edges $\left( i,j\right) $ for $1\leq i<j\leq N$ carrying submodular edge-lengths. Finding a shortest $M$-link path from $1$ to $N$ in $G$ for any given $1<M<N-1$ has many applications. In this paper, we give a contract-and-conquer algorithm for this problem which runs in $O\left( \sqrt{NM\left( N-M\right) \log\left( N-M\right) }\right) $ time and $O\left( N\right) $ space. It is the first $o\left( NM\right) $-time algorithm with linear space complexity, and its time complexity decreases with $M$ when $M\geq N/2$. In contrast, all previous strongly polynomial algorithms have running time growing with $M$. For both $O\left( poly\left( \log N\right) \right) $ and $N-O\left( poly\left( \log N\right) \right) $ regimes of $M$, our algorithm has running time $O\left( N\cdot poly\left( \log N\right) \right) $, which partially answers an open question rased in \cite{AST94} affirmatively.

Finding a Shortest $M$-link Path in a Monge Directed Acyclic Graph

TL;DR

This work addresses finding a shortest -link path in a Monge DAG with submodular edge lengths, a problem with broad algorithmic implications. The authors introduce a contract-and-conquer framework that blends a parsimonious dynamic programming, parametric shortest paths, and iterative contractions to achieve time and space, the first sub- time algorithm with linear space. Their analysis covers both polylogarithmic and near-extremal regimes, delivering performance in the and cases and offering a clear comparison to prior strongly polynomial methods. The approach is conceptually simple, relies on elementary decreasing-and-conquer ideas, and opens avenues for hybrid strategies that adapt to the relative size of and in practice.

Abstract

A Monge directed acyclic graph (DAG) on the nodes has edges for carrying submodular edge-lengths. Finding a shortest -link path from to in for any given has many applications. In this paper, we give a contract-and-conquer algorithm for this problem which runs in time and space. It is the first -time algorithm with linear space complexity, and its time complexity decreases with when . In contrast, all previous strongly polynomial algorithms have running time growing with . For both and regimes of , our algorithm has running time , which partially answers an open question rased in \cite{AST94} affirmatively.
Paper Structure (7 sections, 11 theorems, 71 equations, 2 figures, 4 tables)

This paper contains 7 sections, 11 theorems, 71 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A Parsimonious Dynamic Programming
  3. Minimal and Maximal Shortest-Path Trees
  4. Parametric Shortest Paths
  5. The $m$-Probe
  6. The Contract-and-Conquer Algorithm
  7. Conclusion

Key Result

Theorem 3.1

The depth of each $n\in\left[ s:N\right]$ in $T_{s}^{\min}$ (resp., $T_{s}^{\max}$) is exactly $d_{s}^{\min}\left( n\right)$ (resp., $d_{s}^{\max}\left( n\right)$). In addition, all nodes with the same depth in $T_{s}^{\min}$ (resp, $T_{s}^{\max}$) are consecutive.

Figures (2)

  • Figure 1: Minimal recurrence dependence.
  • Figure 2: Path swapping.

Theorems & Definitions (11)

  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Corollary 4.4
  • Theorem 5.1
  • Lemma 5.2
  • ...and 1 more