Log-Compressed Depth Representation

• Log-compressed depth representation is a technique that reduces effective tree depth by identifying and compacting recurring subtree structures.
• Minimal DAG compaction transforms trees into a succinct form with Theta(n/ln n) nodes through analytic methods like generating functions and singularity analysis.
• The Height Compression Theorem converts deep computation trees into binary trees of logarithmic depth, enabling space-efficient simulations with reduced stack usage.

A log-compressed depth representation is a strategy for reducing the effective depth and size of structured objects—primarily trees and computation trees—by leveraging recurring structural patterns and balanced tree transformations. This approach is of central importance in data compression, succinct data structure design, and space-efficient simulation of computation, with rigorous analysis rooted in combinatorics and algorithmic theory. Two primary instantiations are: minimal directed acyclic graph (DAG) representations via fringe-subtree compaction for random logarithmic-depth trees (Bodini et al., 2020), and tree height compression techniques for canonical computation trees (&&&1&&&).

1. Fringe-Subtree Compaction and Minimal DAGs

A canonical realization of log-compressed depth is the compaction of a rooted tree $T$ of size $n$ by collapsing duplicate fringe (i.e., root-to-leaf) subtrees into a single shared copy. The procedure traverses the tree (commonly postorder), mapping each unique (unlabeled) subtree shape to a representative node. Upon encountering a previously seen shape, further expansion is halted, and a pointer is set to the existing instance. The resulting structure is a minimal DAG whose nodes represent the distinct fringe subtrees of $T$.

Among shallow trees, particularly those with depth $O(\log n)$—such as recursive trees and plane binary increasing trees—the frequency of small, repeating subtrees ensures that the count of unique shapes is far less than $n$. For these models, the expected number $E[X_n]$ of nodes in the compacted DAG is $O(n/\ln n)$, with lower bounds $\Omega(\sqrt{n})$, and under certain conjectures, the tight bound $E[X_n]=\Theta(n/\ln n)$. This stands in contrast to simply generated tree models where the compacted size is $O(n/\sqrt{\ln n})$ (Bodini et al., 2020).

2. Generating Function Framework and Analytic Bounds

Quantitative analysis rests on exponential generating functions (EGFs) describing tree families. For recursive trees, $T(z) = \int_0^z \exp(T(v)) \, dv$, yielding $T(z) = -\ln(1-z)$, $T_n=(n-1)!$. For plane binary increasing trees, $T(z) = \int_0^z (1+T(v))^2 dv$, or $T(z) = z/(1-z)$, $T_n=n!2^{n-1}$.

To determine the distinct shape count, the EGF $S_t(z)$ of trees lacking a fixed forbidden shape $t$ is analyzed. The recursions for $S_t(z)$ remove the monomial contribution $P_t(z)=\ell(t) z^k/k!$, where $\ell(t)$ is the number of increasing labelings of $t$. Solutions involve singularity analysis: if $\rho$ is the dominant singularity of $T(z)$, and $\tilde{\rho}_t>\rho$ is that of $S_t(z)$, then $\tilde{\rho}_t - 1 \sim C w(t)/k$ ($w(t)=\ell(t)/k!$) for recursive trees and $D w(t)/k^2$ for binary trees. Probabilities are extracted via coefficient comparison and summed over all shapes $t$ up to size $n$, partitioned at $k < \log n$ versus $k \geq \log n$, culminating in $E[X_n]=\Theta(n/\ln n)$ (Bodini et al., 2020).

3. Height Compression Theorem and Computation Trees

Another avenue for log-compressed depth is the Height Compression Theorem, which reshapes unbalanced computation trees arising in space-bounded simulation. Specifically, for a deterministic multitape Turing machine running in time $t$, any left-deep computation tree (of height $\Theta(T),\, T = \lceil t/b \rceil$) can be transformed in logspace to a binary tree $\mathcal{T}'$ of depth $O(\log T)$, while retaining the ability to perform block-size $O(b)$ window replays at the leaf level and $O(1)$ workspace per internal node.

The transformation relies on balanced midpoint recursion—splitting intervals at their centroid—ensuring that for any root-leaf path, the number of simultaneously live tokens or intervals (i.e., stack depth) is bounded by $O(\log T)=O(\log (t/b))$. Path bookkeeping is achieved via constant-size per-level tokens, entirely avoiding the need for wide counters or per-level index storage. Consequently, the space bound for stack and workspace is

$S(b) = O(b) + O(\log (t/b))$

which is minimized for $b = \Theta(\sqrt{t})$ at $S(b) = O(\sqrt{t})$ (Nye, 20 Aug 2025).

4. Prototype Implementations and Empirical Observations

Empirical validation of minimal DAG compaction has been performed for plane binary search trees. Starting from a BST of $n$ distinct keys, node labels are erased and repeated shapes are compacted into a minimal DAG, with each node retaining a representative label plus a short index list (e.g., in-order fill) sufficient to support lookups. For $n$ up to $20\,000$, experimental results show:

• Memory size ratio: compacted/original size empirically decays $\alpha/\ln n$, typically in the $0.4$–$0.5$ range.
• Search time ratio: the number of key comparisons remains identical; overall runtime increases by a factor of $\approx 1.2$, attributed to additional integer-arithmetic per lookup.

This suggests that log-compressed depth representations achieve asymptotically significant compaction with minimal impact on access complexity (Bodini et al., 2020).

5. Generalized Log-Compression Strategies

A general template for log-compression emerges:

1. Identify tree models of polylogarithmic (notably logarithmic) depth and abundant fringe pattern repetition.
2. Express their shape counts analytically with generating functions and forbidden-shape perturbation.
3. Use singularity analysis to bound the shift in dominant singularities and requisite absence probabilities.
4. Aggregate across all candidate shapes and size regimes to establish expected minimal DAG size $\Theta(n/\ln n)$.
5. Implement compaction in $O(n)$ time, showing query complexity is preserved (typically $O(\log n)$ access per operation, with $O(1)$ or $O(\log n)$ overhead).

This methodology extends to a variety of balanced or nearly balanced tree families prevalent in practice (AVL, red-black, treaps, heap-ordered) and is anticipated to yield similar log-compressed representations and storage gains (Bodini et al., 2020). A plausible implication is that by pairing such structural compression with label compression (arithmetic or grammar-based), storage could approach information-theoretic lower bounds for labeled trees of logarithmic depth.

6. Implications for Space-Bounded Computation and Circuit Complexity

The log-compressed depth representation underpins new bounds for space-efficient simulation. The Height Compression Theorem delivers simulation of time $t$ with $O(\sqrt{t})$ space, robust to standard computation models and uniform in $O(\log t)$ space. This has several consequences:

• Branching program upper bounds of $2^{O(\sqrt{s})}$ for bounded-fan-in circuits of size $s$.
• Quadratic time lower bounds for $\SPACE[n]$-complete problems.
• $O(\sqrt{t})$-space certifying interpreters.

With suitable locality assumptions, similar compression extends to multidimensional geometric models (Nye, 20 Aug 2025). This signifies that log-compressed representations serve both as compact data structures and as foundational tools for space-efficient algorithm and complexity theory.

7. Comparative Summary of Key Log-Compressed Depth Constructs

Construct/Model	Depth After Compression	Space Complexity	Analytical Framework
Fringe-subtree compaction (trees)	$O(\log n)$	$\Theta(n/\ln n)$	Generating functions, singularity
Height compression (computation trees)	$O(\log (t/b))$	$O(b + \log (t/b))$	Balanced recursion, token scheme

The approaches delineated efficiently compress depth via structural redundancy (compact DAGs) or balanced decomposition (height-balanced trees), with rigorous analytic methodologies underpinning their bounds and broad applicability across data structures and computational paradigms (Bodini et al., 2020, Nye, 20 Aug 2025).