We study learning-augmented binary search trees (BSTs) via Treaps with carefully designed priorities. The result is a simple search tree in which the depth of each item $x$ is determined by its predicted weight $w_x$. Specifically, each item $x$ is assigned a composite priority of $-\lfloor\log\log(1/w_x)\rfloor + U(0, 1)$ where $U(0, 1)$ is the uniform random variable. By choosing $w_x$ as the relative frequency of $x$, the resulting search trees achieve static optimality. This approach generalizes the recent learning-augmented BSTs [Lin-Luo-Woodruff ICML`22], which only work for Zipfian distributions, by extending them to arbitrary input distributions. Furthermore, we demonstrate that our method can be generalized to a B-Tree data structure using the B-Treap approach [Golovin ICALP'09]. Our search trees are also capable of leveraging localities in the access sequence through online self-reorganization, thereby achieving the working-set property. Additionally, they are robust to prediction errors and support dynamic operations, such as insertions, deletions, and prediction updates. We complement our analysis with an empirical study, demonstrating that our method outperforms prior work and classic data structures.

Cross-domain recommendation systems face the challenge of integrating fine-grained user and item relationships across various product domains.To address this, we introduce RankGraph, a scalable graph learning framework designed to serve as a core component in recommendation foundation models (FMs).By constructing and leveraging graphs composed of heterogeneous nodes and edges across multiple products, RankGraph enables the integration of complex relationships between users, posts, ads, and other entities.Our framework employs a GPU-accelerated Graph Neural Network and contrastive learning, allowing for dynamic extraction of subgraphs such as item-item and user-user graphs to support similarity-based retrieval and real-time clustering.Furthermore, RankGraph integrates graph-based pretrained representations as contextual tokens into FM sequence models, enriching them with structured relational knowledge.RankGraph has demonstrated improvements in click (+0.92%) and conversion rates (+2.82%) in online A/B tests, showcasing its effectiveness in cross-domain recommendation scenarios.

We present an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in \(m^{1+o(1)}\) time. Our algorithm builds the flow through a sequence of \(m^{1+o(1)}\) approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized \(m^{o(1)}\) time using a new dynamic graph data structure. Our framework extends to algorithms running in \(m^{1+o(1)}\) time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p -norm flows, and p -norm isotonic regression on arbitrary directed acyclic graphs.

We give the first almost-linear total time algorithm for deciding if a flow of cost at most$F$still exists in a directed graph, with edge costs and capacities, undergoing decremental updates, i.e., edge deletions, capacity decreases, and cost increases. This implies almost-linear time algorithms for approximating the minimum-cost flow value and s-t distance on such decremental graphs. Our framework additionally allows us to maintain decremental strongly connected components in almost-linear time deterministically. These algorithms also improve over the current best known runtimes for statically computing minimum-cost flow, in both the randomized and deterministic settings. We obtain our algorithms by taking the dual perspective, which yields cut-based algorithms. More precisely, our algorithm computes the flow via a sequence of$m^{1+o(1)}$-dynamic min-ratio cut problems, the dual analog of the dynamic min-ratio cycle problem that underlies recent fast algorithms for minimum-cost flow. Our main technical contribution is a new data structure that returns an approximately optimal min-ratio cut in amortized$m^{o(1)}$time by maintaining a tree-cut sparsifier. This is achieved by devising a new algorithm to maintain the dynamic expander hierarchy of [$\text{Goranci-Racke-}$SaranurakTan, SODA 2021] that also works in capacitated graphs. All our algorithms are deterministc, though they can be sped up further using randomized techniques while still working against an adaptive adversary.

The multicommodity flow problem is a classic problem in network flow and combinatorial optimization, with applications in transportation, communication, logistics, and supply chain management, etc. Existing algorithms often focus on low-accuracy approximate solutions, while high-accuracy algorithms typically rely on general linear program solvers. In this paper, we present efficient high-accuracy algorithms for a broad family of multicommodity flow problems on undirected graphs, demonstrating improved running times compared to general linear program solvers. Our main result shows that we can solve the 𝓁_{q, p}-norm multicommodity flow problem to a (1 + ε) approximation in time O_{q, p}(m^{1+o(1)} k² log(1/ε)), where k is the number of commodities, and O_{q, p}(⋅) hides constants depending only on q or p. As q and p approach to 1 and ∞ respectively, 𝓁_{q, p}-norm flow tends to maximum concurrent flow. We introduce the first iterative refinement framework for 𝓁_{q, p}-norm minimization problems, which reduces the problem to solving a series of decomposable residual problems. In the case of k-commodity flow, each residual problem can be decomposed into k single commodity convex flow problems, each of which can be solved in almost-linear time. As many classical variants of multicommodity flows were shown to be complete for linear programs in the high-accuracy regime [Ding-Kyng-Zhang, ICALP'22], our result provides new directions for studying more efficient high-accuracy multicommodity flow algorithms.

We provide an algorithm which, with high probability, maintains a (1 — ɛ)-approximate maximum flow on an undirected graph undergoing m-edge additions in amortized mo(1)ɛ-3 time per update. To obtain this result, we provide a more general algorithm that solves what we call the incremental, thresholded, p-norm flow problem that asks to determine the first edge-insertion in an undirected graph that causes the minimum ℓp-norm flow to decrease below a given threshold in value. Since we solve this thresholded problem, our data structure succeeds against an adaptive adversary that can only see the data structure's output. Furthermore, since our algorithm holds for p = 2, we obtain improved algorithms for dynamically maintaining the effective resistance between a pair of vertices in an undirected graph undergoing edge insertions.

We give the first almost-linear time algorithms for several problems in incremental graphs including cycle detection, strongly connected component maintenance, s-t shortest path, maximum flow, and minimum-cost flow. To solve these problems, we give a deterministic data structure that returns a mo(1)-approximate minimum-ratio cycle in fully dynamic graphs in amortized mo(1) time per update. Combining this with the interior point method framework of Brand-Liu-Sidford (STOC 2023) gives the first almost-linear time algorithm for deciding the first update in an incremental graph after which the cost of the minimum-cost flow attains value at most some given threshold F. By rather direct reductions to minimum-cost flow, we are then able to solve the problems in incremental graphs mentioned above.

We give a deterministic $m^{1+o(1)}$ time algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities. As a consequence, we obtain the first running time improvement for deterministic algorithms that compute maximum-flow in graphs with polynomial bounded capacities since the work of Goldberg-Rao [J.ACM ’98].Our algorithm builds on the framework of Chen-Kyng-Liu-Peng-Gutenberg-Sachdeva [FOCS ’22] that computes an optimal flow by computing a sequence of $m^{1+o(1)}$-approximate undirected minimum-ratio cycles. We develop a deterministic dynamic graph data-structure to compute such a sequence of minimum-ratio cycles in an amortized $m^{o(1)}$ time per edge update. Our key technical contributions are deterministic analogues of the vertex sparsification and edge sparsification components of the data-structure from Chen et al. For the vertex sparsification component, we give a method to avoid the randomness in Chen et al. which involved sampling random trees to recurse on. For the edge sparsification component, we design a deterministic algorithm that maintains an embedding of a dynamic graph into a sparse spanner. We also show how our dynamic spanner can be applied to give a deterministic data structure that maintains a fully dynamic low-stretch spanning tree on graphs with polynomially bounded edge lengths, with subpolynomial average stretch and subpolynomial amortized time per edge update.

We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in $m^{1+o(1)}$ time. Our algorithm builds the flow through a sequence of $m^{1+o(1)}$ approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized $m^{o(1)}$ time using a new dynamic graph data structure. Our framework extends to algorithms running in $m^{1+o(1)}$ time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and p-norm isotonic regression on arbitrary directed acyclic graphs.

Diffusion is a fundamental graph procedure and has been a basic building block in a wide range of theoretical and empirical applications such as graph partitioning and semi-supervised learning on graphs. In this paper, we study computationally efficient diffusion primitives beyond random walk. We design an near-linear time randomized algorithm for the 2-norm flow diffusion problem, a recently proposed diffusion model based on network flow with demonstrated graph clustering related applications both in theory and in practice. Examples include finding locally-biased low conductance cuts. Using a known connection between the optimal dual solution of the flow diffusion problem and the local cut structure, our algorithm gives an alternative approach for finding such cuts in nearly linear time. From a technical point of view, our algorithm contributes a novel way of dealing with inequality constraints in graph optimization problems. It adapts the high-level algorithmic framework of nearly linear time Laplacian system solvers, but requires several new tools: vertex elimination under constraints, a new family of graph ultra-sparsifiers, and ac-celerated proximal gradient methods with inexact proximal mapping computation. See https://arxiv.org/abs/2105.14629 for the full version of this paper.

We present a general framework of designing efficient dynamic approximate algorithms for optimization problems on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1)A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in ~O(n2/3)11The ~O(·) notation is used in this paper to hide poly-logarithmic factors. amortized time against an oblivious adversary, and ~O(m3/4) time against an adaptive adversary. (2)An incremental data structure that maintains O(1) - approximate shortest path in no(1)time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in ~O(n2/3+o(1)) amortized time per operation. (3)A fully-dynamic algorithm that approximates all-pair effective resistance up to an ( 1+ε) factor in ~O(n2/3+o(1)ε-O(1)) amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The O(1)-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM '05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC '19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy. See https://arxiv.org/pdf/2005.02368.pdf for the full version of this paper.