arXiv subjects

Yaniv Sadeh

Publications and source records attributed to Yaniv Sadeh.

8 recordsLinked to original sources

arXiv2026

Beyond Vizing Chains: Improved Recourse in Dynamic Edge Coloring

We study the maintenance of a $(Δ+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $Δ$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates. We present a new technique based on an object which we call a \emph{shift-tree}. This object tracks multiple possible recolorings of $G$ and enables us to maintain a proper coloring with small recourse in polynomial time. We shift colors over a path of edges, but unlike many other algorithms, we do not use \emph{fans} and \emph{alternating bicolored paths}. We combine the shift-tree with additional techniques to obtain an algorithm with a \emph{tight} recourse of $O\big( \frac{\log n}{\log \frac{Δ+C}{Δ-C}}\big)$ for all $C \ge 0.62Δ$ where $Δ-C = O(n^{1-δ})$. Our algorithm is the first deterministic algorithm to establish tight bounds for large palettes, and the first to do so when $Δ-C=o(Δ)$. This result settles the theoretical complexity of the recourse for large palettes. Furthermore, we believe that viewing the possible shifts as a tree can lead to similar tree-based techniques that extend to lower values of $C$, and to improved update times. A second application is to graphs with low arboricity $α$. Previous works [BCPS24, CRV24] achieve $O(ε^{-1}\log n)$ recourse per update with $C\ge (4+ε)α$, and we improve by achieving the same recourse while only requiring $C \ge (2+ε)α- 1$. This result is $Δ$-adaptive, i.e., it uses $Δ_t+C$ colors where $Δ_t$ is the current maximum degree. Trying to understand the limitations of our technique, and shift-based algorithms in general, we show a separation between the recourse achievable by algorithms that only shift colors along a path, and more general algorithms such as ones using the Nibbling Method [BGW21, BCPS24].

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arXiv2026

Dynamic Edge Coloring of Forests

In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $\Delta$ with at most $\Delta+c$ colors, under edge updates. A prominent objective is to minimize the \emph{recourse}, namely the number of edges that are recolored. We study this problem on forests, arguably the simplest graph class that already captures much of the complexity of the problem. We consider both the \emph{incremental} model, where edges are only inserted and the \emph{fully dynamic} model where edges may also be deleted. In the deterministic setting, we focus on the natural greedy algorithm. We show that it achieves $O(\frac{1}{c + \sqrt{\Delta}})$ amortized recourse in the incremental model, and that this is tight up to tie-breaking. In contrast, in a fully dynamic forest, greedy can be forced to have $\Omega(\log_\Delta n)$ amortized recourse. To partially overcome this limitation of greedy within the deterministic setting, we give an optimal non-greedy algorithm with $O(1)$ amortized recourse for \emph{rooted} fully dynamic forests and $c=\Delta-2$. In the randomized setting, we give a natural distribution-maintaining algorithm. In the incremental model, it achieves $\Theta(\frac{1}{\Delta})$ expected amortized recourse, and we show that this is optimal for every constant $c$. In the fully dynamic model, the same algorithm achieves $\Theta(\min \{ \frac{\Delta}{c}, \log_{\Delta} n \})$ expected recourse for $c > 0$, and $\Theta(\log_{\Delta} n)$ for $c = 0$. We show that this is optimal for $c = 0$, and prove an $\Omega(1)$ lower bound for every constant $c$.

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arXiv2025

Search Trees on Trees via LP

We consider the problem of computing optimal search trees on trees (STTs). STTs generalize binary search trees (BSTs) in which we search nodes in a path (linear order) to search trees that facilitate search over general tree topologies. Golinsky proposed a linear programming (LP) relaxation of the problem of computing an optimal static STT over a given tree topology. He used this LP formulation to compute an STT that is a $2$-approximation to an optimal STT, and conjectured that it is, in fact, an extended formulation of the convex-hull of all depths-vectors of STTs, and thus always gives an optimal solution. In this work we study this LP approach further. We show that the conjecture is false and that Golinsky's LP does not always give an optimal solution. To show this we use what we call the ``normals method''. We use this method to enumerate over vertices of Golinsky's polytope for all tree topologies of no more than 8 nodes. We give a lower bound on the integrality gap of the LP and on the approximation ratio of Golinsky's rounding method. We further enumerate several research directions that can lead to the resolution of the question whether one can compute an optimal STT in polynomial time.

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arXiv2024

Caching Connections in Matchings

Motivated by the desire to utilize a limited number of configurable optical switches by recent advances in Software Defined Networks (SDNs), we define an online problem which we call the Caching in Matchings problem. This problem has a natural combinatorial structure and therefore may find additional applications in theory and practice. In the Caching in Matchings problem our cache consists of $k$ matchings of connections between servers that form a bipartite graph. To cache a connection we insert it into one of the $k$ matchings possibly evicting at most two other connections from this matching. This problem resembles the problem known as Connection Caching, where we also cache connections but our only restriction is that they form a graph with bounded degree $k$. Our results show a somewhat surprising qualitative separation between the problems: The competitive ratio of any online algorithm for caching in matchings must depend on the size of the graph. Specifically, we give a deterministic $O(nk)$ competitive and randomized $O(n \log k)$ competitive algorithms for caching in matchings, where $n$ is the number of servers and $k$ is the number of matchings. We also show that the competitive ratio of any deterministic algorithm is $Ω(\max(\frac{n}{k},k))$ and of any randomized algorithm is $Ω(\log \frac{n}{k^2 \log k} \cdot \log k)$. In particular, the lower bound for randomized algorithms is $Ω(\log n)$ regardless of $k$, and can be as high as $Ω(\log^2 n)$ if $k=n^{1/3}$, for example. We also show that if we allow the algorithm to use at least $2k-1$ matchings compared to $k$ used by the optimum then we match the competitive ratios of connection catching which are independent of $n$. Interestingly, we also show that even a single extra matching for the algorithm allows to get substantially better bounds.

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arXiv2023

Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm

Binary search trees (BSTs) are one of the most basic and widely used data structures. The best static tree for serving a sequence of queries (searches) can be computed by dynamic programming. In contrast, when the BSTs are allowed to be dynamic (i.e. change by rotations between searches), we still do not know how to compute the optimal algorithm (OPT) for a given sequence. One of the candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant factor is known by the name Greedy Future (GF). In an equivalent geometric way of representing queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG). Most of the results on GF are obtained using the geometric model and the study of GG. Despite this intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is $\frac{4}{3}$. Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only linear in the number of queries. In this paper we prove a lower bound of $2$ on the competitive ratio of GF, and we prove that the additive gap between the cost of GF and OPT can be $Ω(m \cdot \log\log n)$ where $n$ is the number of items in the tree and $m$ is the number of queries.

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arXiv2022

Codes for Load Balancing in TCAMs: Size Analysis

Traffic splitting is a required functionality in networks, for example for load balancing over paths or servers, or by the source's access restrictions. The capacities of the servers (or the number of users with particular access restrictions) determine the sizes of the parts into which traffic should be split. A recent approach implements traffic splitting within the ternary content addressable memory (TCAM), which is often available in switches. It is important to reduce the amount of memory allocated for this task since TCAMs are power consuming and are often also required for other tasks such as classification and routing. Recent works suggested algorithms to compute a smallest implementation of a given partition in the longest prefix match (LPM) model. In this paper we analyze properties of such minimal representations and prove lower and upper bounds on their size. The upper bounds hold for general TCAMs, and we also prove an additional lower-bound for general TCAMs. We also analyze the expected size of a representation, for uniformly random ordered partitions. We show that the expected representation size of a random partition is at least half the size for the worst-case partition, and is linear in the number of parts and in the logarithm of the size of the address space.

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arXiv2022

On Ranges and Partitions in Optimal TCAMs

Traffic splitting is a required functionality in networks, for example for load balancing over paths or servers, or by the source's access restrictions. The capacities of the servers (or the number of users with particular access restrictions) determine the sizes of the parts into which traffic should be split. A recent approach implements traffic splitting within the ternary content addressable memory (TCAM), which is often available in switches. It is important to reduce the amount of memory allocated for this task since TCAMs are power consuming and are often also required for other tasks such as classification and routing. In the longest-prefix model (LPM), Draves et al. (INFOCOM 1999) find a minimal representation of a function, and Sadeh et al. (INFOCOM 2019) find a minimal representation of a partition. In certain situations, range-functions are of special interest, that is, all the addresses with the same target, or action, are consecutive. In this paper we show that minimizing the amount of TCAM entries to represent a partition comes at the cost of fragmentation, such that for some partitions some actions must have multiple ranges. Then, we also study the case where each target must have a single segment of addresses.

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arXiv2021

Optimal Weighted Load Balancing in TCAMs

Traffic splitting is a required functionality in networks, for example for load balancing over multiple paths or among different servers. The capacities of the servers determine the partition by which traffic should be split. A recent approach implements traffic splitting within the ternary content addressable memory (TCAM), which is often available in switches. It is important to reduce the amount of memory allocated for this task since TCAMs are power consuming and are often also required for other tasks such as classification and routing. Previous work showed how to compute the smallest prefix-matching TCAM necessary to implement a given partition exactly. In this paper we solve the more practical case, where at most $n$ prefix-matching TCAM rules are available, restricting the ability to implement exactly the desired partition. We give simple and efficient algorithms to find $n$ rules that generate a partition closest in $L_\infty$ to the desired one. We do the same for a one-sided version of $L_\infty$ which equals to the maximum overload on a server and for a relative version of it. We use our algorithms to evaluate how the expected error changes as a function of the number of rules, the number of servers, and the width of the TCAM.

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