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Massively Parallel Algorithms for Huffman Coding

Masoud Seddighin, Saeed Seddighin

arXiv:2610.09157v1cs.DS

TL;DR

Huffman coding is useful in data-at-scale settings, but straightforward MPC implementations trade local memory against round complexity. This paper gives a different algorithm, achieving O_ϵ(log log n) rounds with eO(n^ϵ) memory per machine and eO(n) total memory and work.

  • Problem

    Huffman coding has straightforward MPC implementations that require either √n local memory or logarithmic rounds, despite its practical relevance to data-at-scale processing.

    Source: §1 · pp. 1–2

  • Method

    The paper uses a different framework to construct an MPC algorithm for Huffman coding.

    Source: §1 · p. 2, §3 · p. 3

  • Results

    O_ϵ(log log n) rounds and eO(n^ϵ) memory per machine, with eO(n) total memory and computation.

    Source: §3 · p. 3

  • Takeaways & Limitations

    The paper presents the first nontrivial MPC algorithm for Huffman coding beyond the straightforward implementations.

    Source: §3 · p. 3

    The reconstruction account assumes the optimal leaf depths are specified and initially takes the target depth as given.

    Source: §3.3 · p. 11, §6 · p. 23

Abstract

from arXiv · show

Huffman coding is one of the oldest and most fundamental problems in computer science. Given a string of length $\TextLength$ over a general alphabet, the goal is to assign a binary codeword to each character so that no codeword is a prefix of another and the total encoded length of the string is minimized. Huffman coding is widely used in practical compression systems, including file compression. As modern datasets continue to grow, it is natural to study whether a Huffman code can be constructed efficiently in the massively parallel computation (\MPC) model. The celebrated Huffman coding algorithm admits two straightforward \MPC implementations: for any constant $\epsilon\in(0,1)$, one uses $O(\TextLength^\epsilon)$ memory per machine but requires $\Theta(\log \TextLength)$ rounds, while the other runs in $O(1)$ rounds but requires $\Theta(\sqrt{\TextLength})$ memory per machine. We give the first nontrivial \MPC algorithm for Huffman coding that bypasses both limitations. For every constant $\epsilon>0$, our algorithm uses $O_\epsilon(\log\log \TextLength)$ rounds and $\softO(\TextLength^\epsilon)$ memory per machine, while its total memory and total computation are $\softO(\TextLength)$. This provides a rare example in which an exact solution to a problem whose classical algorithm is sequential in nature can be obtained in a sublogarithmic number of \MPC rounds.

1 Introduction

Huffman coding matters for practical data-at-scale systems and presents a challenge for MPC. The introduction contrasts straightforward memory–round tradeoffs with an approach using O_ϵ(log log n) rounds and eO(n^ϵ) memory per machine.

1 Introduction

  • Huffman coding greedily merges the two smallest frequencies until one tree remains, while MPC is motivated by data-at-scale practice and the difficulty of parallelizing sequential combinatorial problems.The resulting tree assigns each character a codeword along its root-to-leaf path.Source: §1 · p. 1
  • The first straightforward approach gathers a compressed frequency representation on one machine, achieving O(1) rounds with O(√n) local memory.It stores counts for frequencies up to √n and explicitly stores larger frequencies.Source: §1 · p. 2
  • The second straightforward approach processes sorted frequencies in dyadic-range phases, using O(log n) rounds, O(n) total memory, and eO(n) total computation.It carries at most one frequency below the current phase range and merges pairs within each range in parallel.Source: §1 · p. 2
  • The new algorithm uses O_ϵ(log log n) rounds and eO(n^ϵ) memory per machine, with eO(n) total work and memory.It is presented as bypassing the barriers of the straightforward implementations.Source: §1 · p. 2
  • For polynomially bounded input frequencies of length m, the approach uses O_ϵ(log log m) rounds and O_ϵ(m^ϵ) memory per machine, with O_ϵ(m^(1+ϵ)) total memory and work.The round and per-machine memory bounds remain as the input changes from raw text to frequencies.Source: §1 · p. 2

2 Preliminaries

The preliminaries define prefix-free binary codes and the encoded-length objective, then state that the main algorithm computes the minimum encoded length.

2 Preliminaries

  • A binary code is prefix-free when no codeword is a prefix of another; this makes concatenated codewords unambiguous to decode.The text gives {0, 10, 11} as prefix-free and {0, 01} as not prefix-free.Source: §2 · p. 3
  • The encoded text length is the sum of each character frequency times its codeword length, and Huffman coding minimizes this quantity.The main algorithm focuses on computing the minimum encoded length, denoted µ(T).Source: §2 · p. 3

3 Our Results

The section presents parallel methods for Huffman coding, combining bounded-ratio merge procedures with reconstruction of a target state of the classic Huffman process. These methods use layered convexity, parallel layer searches, and fractional-knapsack cutoffs.

3 Our Results

  • For every constant ε > 0, the algorithm computes µ(T) in Oε(log log n) rounds with eO(n^ε) memory per machine and eO(n) total memory and computation.The same algorithm can also construct the codewords, though this section focuses on computing µ(T).Source: §3 · p. 3
  • The general algorithm repeatedly solves a maximal bounded-ratio frequency prefix, reconstructs the corresponding Huffman state without simulating merges, and continues until all frequencies are processed.It takes O(log_λ n) iterations; choosing λ = n^Θ(ε) adds only an Oε(1) factor to the round complexity.Source: §3 · pp. 4–6

3.1 Bounded-Ratio Huffman Coding

  • For bounded-ratio inputs, optimal depths have a short range; frequency blocks with ratio at most φ can be solved independently in O(1) MPC rounds and combined in a parallel hierarchy.The hierarchy has O(log(2 + log λ)) levels, and its root yields the optimal encoded length.Source: §3.1 · pp. 6–7

3.2 Fast Merging by Residue Convex Decomposition

  • For each feasible parent budget, the merge split-cost function is convex within each residue class modulo d = 2u−ℓ−1, enabling independent parallel searches for class minima.The function need not be convex across consecutive budget divisions, but layered convexity lets the algorithm find the overall minimum without enumerating every split.Source: §3.2 · pp. 8–9
  • Residue decomposition makes the merge search layered convex: fixing choices below width d leaves the t most valuable width-d items for budget r + td, enabling independent layer searches.Probing η evenly spaced positions per layer reduces the search range by Θ(η) per round; a merge takes O(1 + log_η min{|A1|, |A2|}) rounds.Source: §3.2 · p. 10
  • Bounded-Ratio Huffman Coding on m frequencies takes O((1 + log_η m) log(2 + log λ)) MPC rounds, with O(ηλ² log_φ²) local memory per machine and eO(mηλ⁴ log_φ²) total memory and computation.These bounds hold for every η ≥ 2.Source: §3.2 · p. 11

3.3 Iterative-State Reconstruction

  • For each target Kraft value, fractional knapsack orders leaves by frequency-to-Kraft-value density, and an optimal integral solution differs by at most 2c leaves per depth group.This yields residual instances of O(c(hmax − hmin)) elements; computing P(r) for each target and taking successive differences recovers the node frequencies.Source: §3.3 · pp. 12–13
  • Iterative-State Reconstruction is exact in constant MPC rounds, with eO(mλ log_φ²) total memory and computation.It uses the bounded-ratio solution's leaf depths to reconstruct the classic Huffman algorithm's state immediately before its first merge involving a frequency greater than τ.Source: §3.3 · pp. 11, 13

4 Bounded-Ratio Huffman Coding

Bounded-Ratio Huffman Coding restricts optimal depths to a bounded range, turns Kraft feasibility into a budget constraint, and solves the resulting optimization with consecutive frequency blocks and a balanced hierarchy. The algorithm computes the optimum and can recover an optimal depth vector.

4 Bounded-Ratio Huffman Coding

  • An optimal tree can restrict every leaf depth to [ℓ, u], where ℓ=⌈log2 m⌉−Δλ and u=⌊log2 m⌋+Δλ, making the Kraft constraint an integer budget.The depth difference is bounded using the growth of node weights along a Huffman path.Source: §4 · pp. 14–15
  • The bounded-range problem assigns each frequency a depth in [ℓ, u] while minimizing encoded length subject to a budget upper bound; at budget 2u this retains the original optimum.The budget contribution of a depth hi is 2^(u−hi).Source: §4 · pp. 15–16
  • Consecutive frequency blocks have ratio at most φ, and their optimum budget tables are combined bottom-up by dividing each parent budget between child tables.Pairwise merges proceed in parallel across a balanced hierarchy, with O(log2 q) stages; the final table entry is the optimum encoded length.Source: §4 · pp. 16, 18
  • Every φ-bounded frequency block has an optimum using at most three consecutive depths, allowing all its budget values to be computed in O(1) MPC rounds.The block construction guarantees a frequency ratio of at most φ, and prefix sums provide costs for consecutive ranges.Source: §4 · p. 16
  • The algorithm stores minimizing budget splits and block witnesses, then follows them down the hierarchy to recover an optimal depth vector.Within each block, the stored three-depth window and counts divide sorted frequencies into consecutive ranges.Source: §4 · pp. 17–18

5 Fast Merging by Layered Convexity

Fast merging must compute the best division of each parent budget between two child frequency sets without scanning every feasible split. The paper proves a residue-wise convexity property and uses parallel searches to find the minimizing divisions.

5 Fast Merging by Layered Convexity

  • Each parent-table value requires minimizing the sum of two child-table costs over feasible divisions of the parent budget.Direct evaluation may require checking O(min{|A1|, |A2|}2^(u−ℓ)) divisions.Source: §5 · p. 19
  • The depth-assignment problem becomes a knapsack over upgrades: each step costs 2^(t−1) Kraft units and saves fi in encoded length, and an optimal choice can be represented by upgrade prefixes.Replacing selected upgrades of a frequency by its smallest upgrades preserves value and does not increase size, yielding a valid depth assignment.Source: §5 · pp. 19–20
  • For every merge and parent budget, the profile Gb is layered convex modulo d=2^(u−ℓ−1): it is convex within each residue class.This is weaker than ordinary convexity, which the profile need not satisfy.Source: §5 · pp. 19, 21
  • Within each residue layer, adjacent-value comparisons locate a minimum, so searching the layers avoids scanning every feasible budget split.The minimum over all feasible divisions is the smallest of the layer minima.Source: §5 · pp. 19, 21
  • Each MPC round probes η evenly spaced positions per active interval, shrinking candidate divisions by Θ(η); all layers and parent budgets are searched in parallel.The search evaluates the remaining candidates directly once intervals reach constant size.Source: §5 · pp. 21–22
  • The merge procedure is applied throughout a balanced hierarchy, and the algorithm returns the optimum value and a corresponding optimal depth vector.Merges at the same level involve disjoint multisets whose sizes sum to m.Source: §5 · p. 22

6 Iterative-State Reconstruction

Iterative-State Reconstruction uses supplied leaf depths to recover the Huffman state at a threshold without simulating merges one by one. It reconstructs weights at a chosen depth from minimum-cost Kraft-value selections, then computes the stopping state.

6 Iterative-State Reconstruction

  • The procedure returns the active weights immediately before a merge involving a weight greater than τ, plus the cost of earlier merges, using the given optimal leaf depths.It avoids the sequential simulation of intervening Huffman merges.Source: §6 · p. 23
  • Pairing consecutive nodes by weight from deeper levels upward defines a unique canonical full binary tree consistent with the supplied depths.Ties are resolved by the stated canonical ordering; the construction specifies a valid Huffman execution.Source: §6 · p. 23
  • For nodes at depth h*, P(r) is the minimum total frequency of leaves with Kraft value r·2^-h*, and consecutive differences recover the sorted node weights.The first r subtrees attain this minimum, so P(r) is the sum of the r lightest node weights.Source: §6 · pp. 24–25
  • Package-Merge computes these values by pairing low-cost items of equal width into packages, repeatedly increasing their width until selecting the cheapest r final items.Each package preserves the total width and cost of the items it replaces.Source: §6 · p. 25, Figure 11 · p. 25

6.1 Computing the values P(r)

  • A fractional-knapsack relaxation narrows each depth group to a small window around its cutoff, leaving a residual instance that is solved exactly with Package-Merge.The relaxation differs from some optimal integral solution on only a bounded number of leaves; all target instances are solved independently and in parallel.Source: §6.1 · pp. 26–27
  • The algorithm chooses the smallest depth whose minimum node weight is at most τ, reconstructs weights there, then merges eligible consecutive pairs; reconstruction takes O(1 + log_λ m) rounds.The deeper-node weight ordering ensures that nodes below the chosen depth are at most τ, while a new merge at that depth cannot participate before stopping.Source: §6.1 · pp. 27–28

7 Putting the Pieces Together

The full algorithm preprocesses small weights, then repeatedly applies bounded-ratio Huffman coding and iterative-state reconstruction to the active weights. For every constant ε > 0, it achieves O_ε(log log n) rounds, eO(n^ε) local memory, and eO(n) total memory and computation.

7 Putting the Pieces Together

  • Small-weight preprocessing leaves an active multiset whose optimum cost, plus the recorded merge cost, equals the original optimum; its second-smallest weight is at least ⌈n^ε⌉ unless at most one weight remains.The preprocessing performs a prefix of a valid Huffman execution and leaves at most n^(1−ε) + 1 active weights.Source: §7 · pp. 29–30
  • The algorithm repeatedly solves a frequency-bounded prefix, reconstructs its remaining active weights, and updates the accumulated cost until the whole list satisfies the ratio bound.The final bounded-ratio call solves the remaining active instance.Source: §7 · pp. 30–31
  • O_ε(log log n) rounds, eO(n^ε) memory per machine, and eO(n) total memory and computation suffice to compute the optimum encoded length.Theorem 3.1 gives these bounds for every constant ε > 0.Source: §7 · p. 31
  • Each replacement preserves the optimum because merges of prefix weights at most τ are also valid for the full list; the second-smallest active weight grows by a factor greater than λ per nonterminal iteration.Since active weights are at most n, the number of iterations is O_ε(1) for the chosen λ.Source: §7 · p. 31
  • O_ε(1) iterations keep total memory and computation at eO(n), while the bounded-ratio subroutine determines the O_ε(log log n) round complexity.Preprocessing and iterative-state reconstruction each take O_ε(1) rounds.Source: §7 · pp. 31–32

8 Recovering the Codewords

The paper recovers codewords by first reconstructing an optimal binary tree from stored merge and budget-split records, then tracing root-to-leaf paths in parallel.

8 Recovering the Codewords

  • Stored budget splits and contracted-merge children recover an optimal binary tree on the original frequencies, and pointer doubling then recovers every root-to-leaf codeword in O(log log n) rounds.The stated maximum tree depth is O(log n), and codeword recovery does not change the asymptotic resource bounds.Source: §8 · p. 32
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