In a recent breakthrough paper, Chi et al. (STOC'22) introduce an $\tilde{O}(n^{\frac{3 + \omega}{2}})$ time algorithm to compute Monotone Min-Plus Product between two square matrices of dimensions $n \times n$ and entries bounded by $O(n)$. This greatly improves upon the previous $\tilde O(n^{\frac{12 + \omega}{5}})$ time algorithm and as a consequence improves bounds for its applications. Several other applications involve Monotone Min-Plus Product between rectangular matrices, and even if Chi et al.'s algorithm seems applicable for the rectangular case, the generalization is not straightforward. In this paper we present a generalization of the algorithm of Chi et al. to solve Monotone Min-Plus Product for rectangular matrices with polynomial bounded values. We next use this faster algorithm to improve running times for the following applications of Rectangular Monotone Min-Plus Product: $M$-bounded Single Source Replacement Path, Batch Range Mode, $k$-Dyck Edit Distance and 2-approximation of All Pairs Shortest Path. We also improve the running time for Unweighted Tree Edit Distance using the algorithm by Chi et al.

Improved Bounds for Rectangular Monotone Min-Plus Product and Applications

Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-\varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {2.2} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+\varepsilon)^{2-o(1)}$ based on $(\min,+)$-convolution hypothesis. - Partition can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {1.25} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/\varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.

Approximating Knapsack and Partition via Dense Subset Sums

We study the problem of approximating the edit distance of two strings in sublinear time, in a setting where one or both string(s) are preprocessed, as initiated by Goldenberg, Rubinstein, Saha (STOC ’20). Speciﬁcally, in the ( k, K ) -gap edit distance problem, the goal is to distinguish whether the edit distance of two strings is at most k or at least K . We obtain the following results: After preprocessing one string in time n 1+ o (1) , we can solve ( k, k · n o (1) )-gap edit distance in time ( n/k + k ) · n o (1) . After preprocessing both strings separately in time n 1+ o (1) , we can solve ( k, k · n o (1) )-gap edit distance in time kn o (1) . Both results improve upon some previously best known result, with respect to either the gap or the query time or the preprocessing time. Our algorithms build on the framework by Andoni, Krauthgamer and Onak (FOCS ’10) and the recent sublinear-time algorithm by Bringmann, Cassis, Fischer and Nakos (STOC ’22). We replace many complicated parts in their algorithm by faster and simpler solutions which exploit the preprocessing.

Improved Sublinear-Time Edit Distance for Preprocessed Strings

A weighted directed graph $G=(V,A,c)$, where $A\subseteq V\times V$ and $c:A\to R$, describes a road network in which an electric car can roam. An arc $uv$ models a road segment connecting the two vertices $u$ and $v$. The cost $c(uv)$ of an arc $uv$ is the amount of energy the car needs to traverse the arc. This amount may be positive, zero or negative. To make the problem realistic, we assume there are no negative cycles. The car has a battery that can store up to $B$ units of energy. It can traverse an arc $uv\in A$ only if it is at $u$ and the charge $b$ in its battery satisfies $b\ge c(uv)$. If it traverses the arc, it reaches $v$ with a charge of $\min(b-c(uv),B)$. Arcs with positive costs deplete the battery, arcs with negative costs charge the battery, but not above its capacity of $B$. Given $s,t\in V$, can the car travel from $s$ to $t$, starting at $s$ with an initial charge $b$, where $0\le b\le B$? If so, what is the maximum charge with which the car can reach $t$? Equivalently, what is the smallest $\delta_{B,b}(s,t)$ such that the car can reach $t$ with a charge of $b-\delta_{B,b}(s,t)$, and which path should the car follow to achieve this? We refer to $\delta_{B,b}(s,t)$ as the energetic cost of traveling from $s$ to $t$. We let $\delta_{B,b}(s,t)=\infty$ if the car cannot travel from $s$ to $t$ starting with an initial charge of $b$. The problem of computing energetic costs is a strict generalization of the standard shortest paths problem. We show that the single-source minimum energetic paths problem can be solved using simple, but subtle, adaptations of the Bellman-Ford and Dijkstra algorithms. To make Dijkstra's algorithm work in the presence of negative arcs, but no negative cycles, we use a variant of the $A^*$ search heuristic. These results are explicit or implicit in some previous papers. We provide a simpler and unified description of these algorithms.

Optimal energetic paths for electric cars

The edit distance between strings classically assigns unit cost to every character insertion, deletion, and substitution, whereas the Hamming distance only allows substitutions. In many real-life scenarios, insertions and deletions (abbreviated indels ) appear frequently but signiﬁcantly less so than substitutions. To model this, we consider substitutions being cheaper than indels, with cost 1 a for a parameter a ≥ 1. This basic variant, denoted ED a , bridges classical edit distance ( a = 1) with Hamming distance ( a → ∞ ), leading to interesting algorithmic challenges: Does the time complexity of computing ED a interpolate between that of Hamming distance (linear time) and edit distance (quadratic time)? What about approximating ED a ? We ﬁrst present a simple deterministic exact algorithm for ED a and further prove that it is near-optimal assuming the Orthogonal Vectors Conjecture. Our main result is a randomized algorithm computing a (1 + (cid:15) )-approximation of ED a ( X, Y ), given strings X, Y of total length n and a bound k ≥ ED a ( X, Y ). For simplicity, let us focus on k ≥ 1 and a constant (cid:15) > 0; then, our algorithm takes ˜ O ( na + ak 3 ) time. Unless a = ˜ O (1), in which case ED a resembles the standard edit distance, and for the most interesting regime of small enough k , this running time is sublinear in n . We also consider a very natural version that asks to ﬁnd a ( k I , k S ) -alignment , i.e., an alignment with at most k I indels and k S substitutions. In this setting, we give an exact algorithm and, more importantly, an ˜ O ( nk I k S + k S k 3 I )-time (1 , 1+ (cid:15) )-bicriteria approximation algorithm. The latter solution is based on the techniques we develop for ED a for a = Θ( k S k I ), and its running time is again sublinear in n whenever k I (cid:28) k S and the overall distance is small enough. These bounds are in stark contrast to unit-cost edit distance, where state-of-the-art algorithms are far from achieving (1 + (cid:15) )-approximation in sublinear time, even for a favorable choice of k .

An Algorithmic Bridge Between Hamming and Levenshtein Distances

. We present new exact and approximation algorithms for 0-1-Knapsack and Unbounded Knapsack: (cid:4) Exact Algorithm for 0-1-Knapsack: 0-1-Knapsack has known algorithms running in time e O ( n + min { n · OPT , n · W, OPT 2 , W 2 } ) [Bellman ’57], where n is the number of items, W is the weight budget, and OPT is the optimal proﬁt. We present an algorithm running in time e O ( n + ( W + OPT) 1 . 5 ). This improves the running time in case n, W, OPT are roughly equal. (cid:4) Exact Algorithm for Unbounded Knapsack: Unbounded Knapsack has known algorithms running in time e O ( n + min { n · p max , n · w max , p 2max , w 2max } ) [Axiotis, Tzamos ’19, Jansen, Rohwedder ’19, Chan, He ’22], where n is the number of items, w max is the largest weight of any item, and p max is the largest proﬁt of any item. We present an algorithm running in time e O ( n + ( p max + w max ) 1 . 5 ), giving a similar improvement as for 0-1-Knapsack. (cid:4) Approximating Unbounded Knapsack with Resource Augmentation: Unbounded Knapsack has a known FPTAS with running time e O (min { n/ε, n + 1 /ε 2 } ) [Jansen, Kraft ’18]. We study weak approximation algorithms, which approximate the optimal proﬁt but are allowed to overshoot the weight constraint (i.e. resource augmentation). We present the ﬁrst approximation scheme for Unbounded Knapsack in this setting, achieving running time e O ( n + 1 /ε 1 . 5 ). Along the way, we also give a simpler FPTAS with lower order improvement in the standard setting. these results matching conditional

Faster Knapsack Algorithms via Bounded Monotone Min-Plus-Convolution

We revisit the task of computing the edit distance in sublinear time. In the (k,K)-gap edit distance problem we are given oracle access to two strings of length n and the task is to distinguish whether their edit distance is at most k or at least K. It has been established by Goldenberg, Krauthgamer and Saha (FOCS ’19), with improvements by Kociumaka and Saha (FOCS ’20), that the (k,k2)-gap problem can be solved in time O(n/k + poly(k)). One of the most natural questions in this line of research is whether the (k,k2)-gap is best-possible for the running time O(n/k + poly(k)). In this work we answer this question by significantly improving the gap. Specifically, we show that in time O(n/k + poly(k)) we can even solve the (k,k1+o(1))-gap problem. This is the first algorithm that breaks the (k,k2)-gap in this running time. Our algorithm is almost optimal in the following sense: In the low distance regime (k ≤ n0.19) our running time becomes O(n/k), which matches a known n/k1+o(1) lower bound for the (k,k1+o(1))-gap problem up to lower order factors. Our result also reveals a surprising similarity of Hamming distance and edit distance in the low distance regime: For both, the (k,k1+o(1))-gap problem has time complexity n/k1± o(1) for small k. In contrast to previous work, which employed a subsampled variant of the Landau-Vishkin algorithm, we instead build upon the algorithm of Andoni, Krauthgamer and Onak (FOCS ’10) which approximates the edit distance in almost-linear time O(n1+ε) within a polylogarithmic factor. We first simplify their approach and then show how to to effectively prune their computation tree in order to obtain a sublinear-time algorithm in the given time bound. Towards that, we use a variety of structural insights on the (local and global) patterns that can emerge during this process and design appropriate property testers to effectively detect these patterns.

/pdf/almost-optimal-sublinear-time-edit-distance-in-the-low-1q90vo2b.pdf

Almost-optimal sublinear-time edit distance in the low distance regime

In this work we revisit the fundamental Single-Source Shortest Paths (SSSP) problem with possibly negative edge weights. A recent breakthrough result by Bernstein, Nanongkai and Wulff-Nilsen established a near-linear $O(m \log^8(n) \log(W))$-time algorithm for negative-weight SSSP, where $W$ is an upper bound on the magnitude of the smallest negative-weight edge. In this work we improve the running time to $O(m \log^2(n) \log(nW) \log\log n)$, which is an improvement by nearly six log-factors. Some of these log-factors are easy to shave (e.g. replacing the priority queue used in Dijkstra's algorithm), while others are significantly more involved (e.g. to find negative cycles we design an algorithm reminiscent of noisy binary search and analyze it with drift analysis). As side results, we obtain an algorithm to compute the minimum cycle mean in the same running time as well as a new construction for computing Low-Diameter Decompositions in directed graphs.

Negative-Weight Single-Source Shortest Paths in Near-Linear Time: Now Faster!

We revisit the classic 0-1-Knapsack problem, in which we are given $n$ items with their weights and profits as well as a weight budget $W$, and the goal is to find a subset of items of total weight at most $W$ that maximizes the total profit. We study pseudopolynomial-time algorithms parameterized by the largest profit of any item $p_{\max}$, and the largest weight of any item $w_{\max}$. Our main result are algorithms for 0-1-Knapsack running in time $\tilde{O}(n\,w_\max\,p_\max^{2/3})$ and $\tilde{O}(n\,p_\max\,w_\max^{2/3})$, improving upon an algorithm in time $O(n\,p_\max\,w_\max)$ by Pisinger [J. Algorithms '99]. In the regime $p_\max \approx w_\max \approx n$ (and $W \approx \mathrm{OPT} \approx n^2$) our algorithms are the first to break the cubic barrier $n^3$. To obtain our result, we give an efficient algorithm to compute the min-plus convolution of near-convex functions. More precisely, we say that a function $f \colon [n] \mapsto \mathbf{Z}$ is $\Delta$-near convex with $\Delta \geq 1$, if there is a convex function $\breve{f}$ such that $\breve{f}(i) \leq f(i) \leq \breve{f}(i) + \Delta$ for every $i$. We design an algorithm computing the min-plus convolution of two $\Delta$-near convex functions in time $\tilde{O}(n\Delta)$. This tool can replace the usage of the prediction technique of Bateni, Hajiaghayi, Seddighin and Stein [STOC '18] in all applications we are aware of, and we believe it has wider applicability.

Alejandro Cassis

Papers

Faster Knapsack Algorithms via Bounded Monotone Min-Plus-Convolution

Almost-optimal sublinear-time edit distance in the low distance regime

Improved Sublinear-Time Edit Distance for Preprocessed Strings

Negative-Weight Single-Source Shortest Paths in Near-Linear Time: Now Faster!

Faster 0-1-Knapsack via Near-Convex Min-Plus-Convolution