Peleg Algorithm Overview

• Peleg Algorithm is a multifaceted term referring to various procedures by David Peleg and collaborators, covering dynamic matching, game theory, and more.
• In dynamic graph algorithms, it efficiently maintains a (1+ε)-approximate matching in low-arboricity graphs with amortized O(log^4 n) update time.
• Its applications extend to inductively generating minimal balanced collections in game theory, robust facility location, client-server coverage, and phased elimination in linear bandits.

In the cited literature, “Peleg algorithm” is not a canonical name for a single procedure. It is used for several distinct algorithms or algorithmic lineages associated with David Peleg and collaborators, spanning dynamic graph matching, cooperative game theory, distributed MST and synchronizers, robust facility location, one-dimensional coverage, and, in one case, an acronymic linear-bandit method named PELEG, short for “Phased Elimination Linear Exploration Game” (Bhattacharya et al., 2021, Mermoud et al., 8 Jul 2025, Elkin, 2017, Ghaffari et al., 2023, Byrka et al., 2010, Biniaz et al., 2018, Zaki et al., 2020).

1. Terminological scope

The expression is field-dependent. In some papers it denotes a specific algorithm due to Peleg or a Peleg coauthorship line; in others it denotes a black-box subroutine or a historical baseline.

Context	Meaning of “Peleg algorithm”	Representative source
Dynamic matching	Peleg–Solomon dynamic $(1+\varepsilon)$-approximate matching on arboricity-$\mathcal{A}$ graphs	(Bhattacharya et al., 2021)
Cooperative game theory	Peleg’s inductive generation of minimal balanced collections	(Mermoud et al., 8 Jul 2025)
Distributed graph algorithms	Garay–Kutten–Peleg MST framework and Peleg-associated synchronizer lineages	(Elkin, 2017, Ghaffari et al., 2023, Goldenfeld, 12 Mar 2026)
Agreement under sleeping/crash models	Recursive crash agreement co-authored by Meir and Peleg	(Meir et al., 3 Feb 2026)
Facility location and geometric optimization	Chechik–Peleg robust fault-tolerant facility location; Lev-Tov–Peleg client-server coverage DP	(Byrka et al., 2010, Biniaz et al., 2018)
Linear bandits	“PELEG” as an acronym, not a surname-based designation	(Zaki et al., 2020)

A common misconception is that the term has a unique meaning. The cited works suggest the opposite: the phrase is shorthand whose precise referent is determined by subfield, coauthorship, and problem domain.

2. Low-arboricity dynamic matching

In dynamic graph algorithms, the relevant object is the Peleg–Solomon result quoted in “Deterministic Rounding of Dynamic Fractional Matchings” (Bhattacharya et al., 2021). The sole guarantee used there is: “Peleg and Solomon have shown how to deterministically maintain a $(1 + \varepsilon)$-approximate maximum matching in $O(\mathcal{A})$ update time in a dynamic graph with arboricity $\le \mathcal{A}$.” In that usage, “near-optimal” means exactly a $(1+\varepsilon)$-approximate maximum matching.

The 2021 rounding framework is built around a dynamically maintained matching-sparsifier $S=(V,E_s)$. Its key guarantees are that there exists a fractional matching $h'$ on $S$ with

$$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$

that for an $\mathcal{A}$0-approximately maximal fractional matching $\mathcal{A}$1 one has

$\mathcal{A}$2

and that

$\mathcal{A}$3

Moreover, every update in $\mathcal{A}$4 or $\mathcal{A}$5, on average, leads to $\mathcal{A}$6 updates in $\mathcal{A}$7, and the sparsifier itself is maintained in $\mathcal{A}$8 amortized update time (Bhattacharya et al., 2021).

Arboricity is used there in the orientation-based sense: $\mathcal{A}$9 means that $(1 + \varepsilon)$0 admits an orientation in which every vertex has out-degree $(1 + \varepsilon)$1. This matters because the Peleg–Solomon routine is density-sensitive: once the framework ensures $(1 + \varepsilon)$2, the black-box call on $(1 + \varepsilon)$3 costs $(1 + \varepsilon)$4 per update to $(1 + \varepsilon)$5. Composing $(1 + \varepsilon)$6 sparsifier updates with $(1 + \varepsilon)$7 update time on the sparsifier yields the $(1 + \varepsilon)$8 amortized bound.

This composition produces the stated dynamic guarantees. In fully dynamic general graphs, the resulting maintained matching $(1 + \varepsilon)$9 satisfies

$O(\mathcal{A})$0

with $O(\mathcal{A})$1 amortized update time. In decremental bipartite graphs, the same framework yields a $O(\mathcal{A})$2-approximation in $O(\mathcal{A})$3 amortized update time. In fully dynamic bipartite graphs, the paper derives a $O(\mathcal{A})$4-approximation for some $O(\mathcal{A})$5 with update time $O(\mathcal{A})$6 (Bhattacharya et al., 2021). The distinctive feature, as explicitly emphasized there, is that a low-arboricity matching algorithm is used as a black-box subroutine inside a general-graph dynamic rounding framework.

3. Peleg’s inductive algorithm for minimal balanced collections

In cooperative game theory, “the Peleg algorithm” denotes Peleg’s inductive method for generating all minimal balanced collections (MBCs) of coalitions on a finite player set $O(\mathcal{A})$7 (Mermoud et al., 8 Jul 2025). A collection $O(\mathcal{A})$8 is balanced if there exist positive balancing weights $O(\mathcal{A})$9 such that

$\le \mathcal{A}$0

Equivalently, if $\le \mathcal{A}$1 is the incidence matrix of $\le \mathcal{A}$2, then

$\le \mathcal{A}$3

Minimal balanced collections are balanced collections with no balanced proper subcollection. The paper states two equivalent characterizations: minimality is equivalent to uniqueness of balancing weights, and MBCs are exactly the supports of vertices of the polytope

$\le \mathcal{A}$4

The algorithm is inductive. Given $\le \mathcal{A}$5, the set of all MBCs on $\le \mathcal{A}$6, it constructs $\le \mathcal{A}$7 for $\le \mathcal{A}$8. The paper formalizes four constructive cases. In Case 1, if $\le \mathcal{A}$9, the new player $(1+\varepsilon)$0 is inserted into the coalitions indexed by $(1+\varepsilon)$1. In Case 2, if $(1+\varepsilon)$2, $(1+\varepsilon)$3 is inserted into those coalitions and the singleton $(1+\varepsilon)$4 is added with weight $(1+\varepsilon)$5. In Case 3, if $(1+\varepsilon)$6, one refines the collection by splitting $(1+\varepsilon)$7 into $(1+\varepsilon)$8 and $(1+\varepsilon)$9. In Case 4, if $S=(V,E_s)$0 is the union of two different MBCs with $S=(V,E_s)$1, the algorithm uses convex interpolation between the two weight systems and selects $S=(V,E_s)$2. The paper’s correctness theorem states that AddNewPlayer generates all minimal balanced collections on $S=(V,E_s)$3 (Mermoud et al., 8 Jul 2025).

The algorithm is not merely structural. Its implementation generated all MBCs up to $S=(V,E_s)$4: $S=(V,E_s)$5 for $S=(V,E_s)$6, $S=(V,E_s)$7 for $S=(V,E_s)$8, $S=(V,E_s)$9 for $h'$0, $h'$1 for $h'$2, $h'$3 for $h'$4, $h'$5 for $h'$6, and $h'$7 for $h'$8. The sequence was added to the OEIS as A355042. For $h'$9, the implemented Peleg algorithm ran in $S$0 seconds, while Avis–Fukuda vertex enumeration took $S$1 seconds (Mermoud et al., 8 Jul 2025).

Its main applications in that paper are game-theoretic. By the Bondareva–Shapley theorem, a TU-game $S$2 has nonempty core iff for every MBC $S$3,

$S$4

Once $S$5 is precomputed offline, core nonemptiness reduces to finitely many scalar tests. In a benchmark with $S$6 and $S$7 random games, the MBC-based test took $S$8 s total versus $S$9 s for a revised simplex LP. The same paper also extends balancedness from coalition collections to balanced sets $$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$0, and uses that second-level notion in the Grabisch–Sudhölter nested balancedness theorem for testing whether the core is a stable set in the sense of von Neumann and Morgenstern (Mermoud et al., 8 Jul 2025).

4. Distributed computing usages

In distributed algorithms, Peleg-associated algorithms form several major lineages. One is the Garay–Kutten–Peleg MST framework. “A Simple Deterministic Distributed MST Algorithm, with Near-Optimal Time and Message Complexities” summarizes the classic Kutten–Peleg algorithm as a two-phase sublinear-time framework: controlled fragment formation, followed by a BFS-tree pipeline that filters candidate inter-fragment edges on the way to the root. Its stated complexities are

$$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$1

time and

$$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$2

messages, and Peleg–Rubinovich showed that distributed MST requires time $$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$3, making the bound essentially time-optimal up to lower-order factors (Elkin, 2017). A later memory study states that the GKP line is time-efficient but not message- or memory-efficient, analyzing it as using $$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$4 memory per node, and contrasts it with a new deterministic MST/PWA algorithm with

$$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$5

time,

$$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$6

messages, and

$$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$7

bits of memory per vertex (Goldenfeld, 12 Mar 2026).

A second lineage is synchronizers. The deterministic synchronizer paper states that “Peleg’s algorithm” commonly evokes sparse-cover-based synchronizers, specifically the Awerbuch–Peleg and Awerbuch–Patt-Shamir–Peleg–Saks line. Its main informal theorem is that any synchronous event-driven algorithm $$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$8 with time $$\mathrm{size}(w) \le (1+\varepsilon)\cdot \mathrm{size}(h'),$$9 and messages $\mathcal{A}$00 can be simulated asynchronously with time $\mathcal{A}$01 and message complexity $\mathcal{A}$02. The explicit deterministic bounds are larger polylogarithmic expressions, including

$\mathcal{A}$03

time and

$\mathcal{A}$04

messages when $\mathcal{A}$05 is not known in advance (Ghaffari et al., 2023). That work also corrects congestion and model-definition issues in the original conference versions.

A third usage appears in energy-efficient agreement. “Recursive Energy Efficient Agreement” studies crash agreement in the synchronous sleeping model and presents a recursive algorithm, co-authored by Meir and Peleg, with $\mathcal{A}$06 active rounds per participant and $\mathcal{A}$07 total rounds. The optimized construction partitions the system into disjoint blocks of size $\mathcal{A}$08, runs the basic recursive crash-agreement routine inside each block in parallel, then performs a short merge in which each block broadcasts its decision and all parties decide

$\mathcal{A}$09

The recursion depth is

$\mathcal{A}$10

so the awake complexity is $\mathcal{A}$11 (Meir et al., 3 Feb 2026).

5. Facility location and one-dimensional coverage

In approximation algorithms, one meaning of “Peleg algorithm” is the Chechik–Peleg algorithm for robust fault-tolerant facility location (RFTFL). In the metric uncapacitated setting with up to $\mathcal{A}$12 facility failures, Chechik and Peleg introduced the problem and gave a $\mathcal{A}$13-approximation for $\mathcal{A}$14 and a $\mathcal{A}$15-approximation for general $\mathcal{A}$16. The later LP-rounding paper improves this to a $\mathcal{A}$17-approximation for adversarial failures and shows that in the oblivious-failures model the expected approximation ratio can be reduced to $\mathcal{A}$18. It also proves that the integrality gap of the natural robust LP-relaxation is at least $\mathcal{A}$19 (Byrka et al., 2010).

A different geometric usage appears in optimization on collinear points. There, the phrase refers to the Lev-Tov–Peleg dynamic program for minimizing the sum of radii in client-server coverage on a line. The input points are partitioned into clients and servers, and the objective is to minimize $\mathcal{A}$20 so that every client is covered by some interval $\mathcal{A}$21. Lev-Tov and Peleg gave an exact $\mathcal{A}$22-time algorithm. The later paper improves this to $\mathcal{A}$23 by shrinking the candidate set of “last disks” in the DP. If $\mathcal{A}$24 is the rightmost client of the current prefix and $\mathcal{A}$25 is the optimal cost on the first $\mathcal{A}$26 points, the faster recurrence keeps only the disks $\mathcal{A}$27 for servers $\mathcal{A}$28, and the disks $\mathcal{A}$29 where $\mathcal{A}$30 and $\mathcal{A}$31 is the first server strictly to the right of the midpoint $\mathcal{A}$32, yielding

$\mathcal{A}$33

The improvement relies on structural properties already used by Lev-Tov and Peleg: every disk in an optimal solution has at least one client on its boundary, and no disk contains the center of any other non-zero disk (Biniaz et al., 2018).

6. PELEG as an acronym in linear bandits

The string “PELEG” also appears as an acronym unrelated to the author name. In “Explicit Best Arm Identification in Linear Bandits Using No-Regret Learners,” PELEG stands for “Phased Elimination Linear Exploration Game” (Zaki et al., 2020). The problem is fixed-confidence best-arm identification in linearly parameterized bandits: given $\mathcal{A}$34, unknown $\mathcal{A}$35, and noisy observations $\mathcal{A}$36, the goal is to identify

$\mathcal{A}$37

with probability at least $\mathcal{A}$38.

The algorithm is phased. In phase $\mathcal{A}$39, it maintains a confidence ellipsoid

$\mathcal{A}$40

It then formulates a zero-sum game whose value is

$\mathcal{A}$41

where $\mathcal{A}$42 is the active arm set. MAX uses Exponential Weights over arms, MIN plays a best response by solving

$\mathcal{A}$43

and the realized pulls are produced by a deterministic tracking rule. At the end of the phase, arm $\mathcal{A}$44 is eliminated if there exists $\mathcal{A}$45 such that

$\mathcal{A}$46

The algorithm is $\mathcal{A}$47-PAC, terminates in at most $\mathcal{A}$48 phases, and achieves an instance-dependent sample complexity

$\mathcal{A}$49

matching the known lower bound order-wise up to logarithmic factors and an extra geometry term $\mathcal{A}$50. Here

$\mathcal{A}$51

This usage is terminologically important because it is purely acronymic: despite the spelling, it is not another surname-based “Peleg algorithm” (Zaki et al., 2020).

Accordingly, the exact meaning of “Peleg algorithm” must be read from context. In dynamic graph algorithms it usually refers to the Peleg–Solomon low-arboricity matching routine; in cooperative game theory it denotes Peleg’s inductive MBC generator; in distributed computing it points to MST or synchronizer lineages with Peleg coauthorship; in facility location and computational geometry it names Chechik–Peleg or Lev-Tov–Peleg baselines; and in linear bandits it is the acronym PELEG rather than a bibliographic eponym.