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High Dimensional Procedural Content Generation

Kaijie Xu, Clark Verbrugge

TL;DR

High-Dimensional PCG is formally introduced: a framework that elevates non-geometric gameplay dimensions to first-class coordinates of a joint state space and encourages a shift in PCG toward general representations and the generation of gameplay-relevant dimensions beyond geometry, paving the way for controllable, verifiable, and extensible level generation.

Abstract

Procedural content generation (PCG) has made substantial progress in shaping static 2D/3D geometry, while most methods treat gameplay mechanics as auxiliary and optimize only over space. We argue that this limits controllability and expressivity, and formally introduce High-Dimensional PCG (HDPCG): a framework that elevates non-geometric gameplay dimensions to first-class coordinates of a joint state space. We instantiate HDPCG along two concrete directions. Direction-Space augments geometry with a discrete layer dimension and validates reachability in 4D (x,y,z,l), enabling unified treatment of 2.5D/3.5D mechanics such as gravity inversion and parallel-world switching. Direction-Time augments geometry with temporal dynamics via time-expanded graphs, capturing action semantics and conflict rules. For each direction, we present three general, practicable algorithms with a shared pipeline of abstract skeleton generation, controlled grounding, high-dimensional validation, and multi-metric evaluation. Large-scale experiments across diverse settings validate the integrity of our problem formulation and the effectiveness of our methods on playability, structure, style, robustness, and efficiency. Beyond quantitative results, Unity-based case studies recreate playable scenarios that accord with our metrics. We hope HDPCG encourages a shift in PCG toward general representations and the generation of gameplay-relevant dimensions beyond geometry, paving the way for controllable, verifiable, and extensible level generation.

High Dimensional Procedural Content Generation

TL;DR

High-Dimensional PCG is formally introduced: a framework that elevates non-geometric gameplay dimensions to first-class coordinates of a joint state space and encourages a shift in PCG toward general representations and the generation of gameplay-relevant dimensions beyond geometry, paving the way for controllable, verifiable, and extensible level generation.

Abstract

Procedural content generation (PCG) has made substantial progress in shaping static 2D/3D geometry, while most methods treat gameplay mechanics as auxiliary and optimize only over space. We argue that this limits controllability and expressivity, and formally introduce High-Dimensional PCG (HDPCG): a framework that elevates non-geometric gameplay dimensions to first-class coordinates of a joint state space. We instantiate HDPCG along two concrete directions. Direction-Space augments geometry with a discrete layer dimension and validates reachability in 4D (x,y,z,l), enabling unified treatment of 2.5D/3.5D mechanics such as gravity inversion and parallel-world switching. Direction-Time augments geometry with temporal dynamics via time-expanded graphs, capturing action semantics and conflict rules. For each direction, we present three general, practicable algorithms with a shared pipeline of abstract skeleton generation, controlled grounding, high-dimensional validation, and multi-metric evaluation. Large-scale experiments across diverse settings validate the integrity of our problem formulation and the effectiveness of our methods on playability, structure, style, robustness, and efficiency. Beyond quantitative results, Unity-based case studies recreate playable scenarios that accord with our metrics. We hope HDPCG encourages a shift in PCG toward general representations and the generation of gameplay-relevant dimensions beyond geometry, paving the way for controllable, verifiable, and extensible level generation.
Paper Structure (59 sections, 16 equations, 19 figures, 6 tables, 1 algorithm)

This paper contains 59 sections, 16 equations, 19 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Method
  4. General method Definition and Pipeline
  5. High‐dimensional level representation:
  6. Attribute-valued occupancy:
  7. Adjacency and costs:
  8. Dimensional-Expanded Graph:
  9. High-dimensional problem definition:
  10. General solution scheme:
  11. Direction-S: Space
  12. Problem Setup and Modeling
  13. Abstract Skeleton and Controlled Geometry
  14. Methods
  15. Method 1: Naive Noise Baseline
  16. ...and 44 more sections

Figures (19)

  • Figure 1: Space quality. S/M: NP-A* $>$ PF-A* $>$ NNB; L: PF-A* $>$ NP-A* $\gg$ NNB. GA boosts all methods; PF-A* best at L.
  • Figure 2: Controllability-density. PF-A* tracks density targets with lowest MAE at L and competitive at S/M; NP-A* exhibits mild systematic overshoot; NNB is insensitive to targets.
  • Figure 3: Controllability-spacing. PF-A* $\approx$ perfect across scales; NP-A* slightly lower; NNB drops with higher targets.
  • Figure 4: 2D combined-geometry view (Space). Blue/red cells are free only on $L_0/L_1$; green are free on both (switch pockets); grey are blocked on both. Start/end are overlaid. This top-down grid visualization corresponds to our 2.5D Space (gravity-shift) instantiation. The corridor layout and pocket placement obey the target density/spacing used in Sec. \ref{['sec:results-space']}.
  • Figure 5: 3D combined-path view (Space). Blue/red points are free voxels on $L_0/L_1$; purple diamonds are planned switch nodes on the successful route. Alternation frequency matches the requested spacing.
  • ...and 14 more figures