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Parallel Algorithm For Finding The Minimum s/t Cut in a Structured 3-Dimensional Proper Order Graph

Shridharan Chandramouli

TL;DR

The paper tackles scalable parallel computation of the $s-t$ mincut/maxflow on large, 3D structured proper order graphs used for seismic horizon segmentation. It introduces two parallel strategies, with a focus on a cache-aware parallel push-relabel algorithm that uses segmentation, processor affinity, and level-synchronized global relabeling to enable concurrent updates without a global queue. A second approach adapts a parallel Boykov–Kolmogorov style with hierarchical merging to exploit segment-level processing. The key contributions are a novel level-synchronized relabeling scheme, a memory-efficient implicit addressing scheme, and extensive scalability results showing strong speedups on large volumes, highlighting practical impact for real-world image-volume segmentation.

Abstract

We present a parallel algorithm for computing the minimum s-t cut in structured 3-dimensional proper order graphs arising from image segmentation problems. Proper order graphs are multi-column structures where vertices are arranged in parallel columns, with each vertex connected to consecutive vertices in adjacent columns. This graph structure naturally arises in surface extraction problems for geological horizon segmentation in seismic imaging volumes. We develop two parallel approaches: a hierarchical merging variant of the Boykov-Kolmogorov algorithm, and a novel parallel push-relabel algorithm with level synchronized global relabeling. Our primary contribution is the push-relabel variant, which partitions the graph into segments along columns with processor affinity, eliminating the need for a global shared queue. We introduce level synchronized global relabeling that enables concurrent label updates while maintaining correctness through barriers at each frontier level.

Parallel Algorithm For Finding The Minimum s/t Cut in a Structured 3-Dimensional Proper Order Graph

TL;DR

The paper tackles scalable parallel computation of the mincut/maxflow on large, 3D structured proper order graphs used for seismic horizon segmentation. It introduces two parallel strategies, with a focus on a cache-aware parallel push-relabel algorithm that uses segmentation, processor affinity, and level-synchronized global relabeling to enable concurrent updates without a global queue. A second approach adapts a parallel Boykov–Kolmogorov style with hierarchical merging to exploit segment-level processing. The key contributions are a novel level-synchronized relabeling scheme, a memory-efficient implicit addressing scheme, and extensive scalability results showing strong speedups on large volumes, highlighting practical impact for real-world image-volume segmentation.

Abstract

We present a parallel algorithm for computing the minimum s-t cut in structured 3-dimensional proper order graphs arising from image segmentation problems. Proper order graphs are multi-column structures where vertices are arranged in parallel columns, with each vertex connected to consecutive vertices in adjacent columns. This graph structure naturally arises in surface extraction problems for geological horizon segmentation in seismic imaging volumes. We develop two parallel approaches: a hierarchical merging variant of the Boykov-Kolmogorov algorithm, and a novel parallel push-relabel algorithm with level synchronized global relabeling. Our primary contribution is the push-relabel variant, which partitions the graph into segments along columns with processor affinity, eliminating the need for a global shared queue. We introduce level synchronized global relabeling that enables concurrent label updates while maintaining correctness through barriers at each frontier level.
Paper Structure (18 sections, 1 theorem, 6 equations, 13 figures, 1 table, 3 algorithms)

This paper contains 18 sections, 1 theorem, 6 equations, 13 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Proper Order Multi-Column Graph Structure
  4. General Maxflow/Mincut Problem Definition
  5. Related Work
  6. The Push-Relabel Algorithm
  7. The Boykov-Kolmogorov Algorithm
  8. General Parallel Maxflow Algorithms
  9. Parallel Maxflow algorithms for Proper Order Graphs
  10. Parallel BK algorithm with hierarchical merging
  11. Parallel Push-Relabel algorithm with Level Synchronized Global Relabeling
  12. Segmentation strategy and vertex processing
  13. Level Synchronized Global Relabeling
  14. Structured Graph and Implicit Addressing Scheme
  15. Termination
  16. ...and 3 more sections

Key Result

Theorem 1.1

For any network $G$, the value of a maximum $s-t$ flow is equal to the capacity of the minimum $s-t$ cut.

Figures (13)

  • Figure 1: Figure shows the proper order edge structure between adjacent columns in the graph. Each node is connected to the immediate node above and below it and a set of vertices in its adjacent column. The number of connections depends on the edge interval. The figure on left has an edge interval of 2 where as the figure on the right has an edge interval of 3.
  • Figure 2: A simplified 3D proper order graph structure
  • Figure 3: Image shows the 3D image volume segmentation strategy for use with the parallel BK algorithm. The segments are partitioned along the columns and slices
  • Figure 4: The red nodes are connected to sink and the black nodes are connected to the source. After merging, any new path found from source to sink has to include the nodes that are adjacent to one another across old segment boundaries and are marked as belong to different terminal nodes.
  • Figure 5: The orange nodes are shared between adjacent segments and needs to be locked before processing. The gray nodes are interior to the segment and need to be locked only when interacting with a shared vertex.
  • ...and 8 more figures

Theorems & Definitions (1)

  • Theorem 1.1