arXiv subjects

Akash Pareek

Publications and source records attributed to Akash Pareek.

6 recordsLinked to original sources

arXiv2025

Counting Patterns in Degenerate Graphs in Constant Space

For a fixed pattern graph, we study the algorithmic complexity of counting homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms into an $n$-vertex, $d$-degenerate host graph. Bressan (Algorithmica, 2021) introduced the notion of DAG treewidth and showed that counting homomorphisms and induced subgraphs can be performed efficiently using dynamic programming that requires polynomial space. In this work, we introduce a new graph parameter, called DAG treedepth, which enables efficient divide-and-conquer algorithms for counting homomorphisms in $d$-degenerate host graphs using only constant space. Bera, Gishboliner, Levanzov, Seshadhri, and Shapira (SODA, 2021) showed that a pattern graph has DAG treewidth one if and only if it contains no induced cycle of length at least six. This induced minor characterization leads to linear-time and linear-space algorithms. Building on this line of work, we derive an induced-minor characterization of graphs with DAG treedepth at most two that uses only constant space. Recently, Paul-Pena and Seshadhri (ICALP, 2025) proved that all pattern graphs on at most nine vertices can be counted in subquadratic time using polynomial space. We show that every pattern graph on at most nine vertices can be counted as an induced subgraph in $O(n^3)$ time using only constant space. Moreover, we show that patterns on at most eleven vertices can be counted in $O(n^2)$ time using polynomial space. Finally, we present a constant-space algorithm for counting induced subgraphs that matches the running time of Bressan algorithm. We further show that, when polynomial space is allowed, homomorphisms, subgraph isomorphisms, and induced subgraph isomorphisms can be counted faster than Bressan algorithm. In addition, we establish several other results related to DAG treewidth and DAG treedepth that may be of independent interest.

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arXiv2025

Ads that Stick: Near-Optimal Ad Optimization through Psychological Behavior Models

Optimizing the timing and frequency of ads is a central problem in digital advertising, with significant economic consequences. Existing scheduling policies rely on simple heuristics, such as uniform spacing and frequency caps, that overlook long-term user interest. However, it is well-known that users' long-term interest and engagement result from the interplay of several psychological effects (Curmei, Haupt, Recht, Hadfield-Menell, ACM CRS, 2022). In this work, we model change in user interest upon showing ads based on three key psychological principles: mere exposure, hedonic adaptation, and operant conditioning. The first two effects are modeled using a concave function of user interest with repeated exposure, while the third effect is modeled using a temporal decay function, which explains the decline in user interest due to overexposure. Under our psychological behavior model, we ask the following question: Given a continuous time interval $T$, how many ads should be shown, and at what times, to maximize the user interest towards the ads? Towards answering this question, we first show that, if the number of displayed ads is fixed, then the optimal ad-schedule only depends on the operant conditioning function. Our main result is a quasi-linear time algorithm that outputs a near-optimal ad-schedule, i.e., the difference in the performance of our schedule and the optimal schedule is exponentially small. Our algorithm leads to significant insights about optimal ad placement and shows that simple heuristics such as uniform spacing are sub-optimal under many natural settings. The optimal number of ads to display, which also depends on the mere exposure and hedonistic adaptation functions, can be found through a simple linear search given the above algorithm. We further support our findings with experimental results, demonstrating that our strategy outperforms various baselines.

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arXiv2025

A Simple PTAS for Weighted $k$-means and Sensor Coverage

Clustering is a fundamental technique in data analysis, with the $k$-means being one of the widely studied objectives due to its simplicity and broad applicability. In many practical scenarios, data points come with associated weights that reflect their importance, frequency, or confidence. Given a weighted point set $P \subset R^d$, where each point $p \in P$ has a positive weight $w_p$, the goal is to compute a set of $k$ centers $C = \{ c_1, c_2, \ldots, c_k \} \subset R^d$ that minimizes the weighted clustering cost: $\Delta_w(P,C) = \sum_{p \in P} w_p \cdot d(p,C)^2$, where $d(p,C)$ denotes the Euclidean distance from $p$ to its nearest center in $C$. Although most existing coreset-based algorithms for $k$-means extend naturally to the weighted setting and provide a PTAS, no prior work has offered a simple, coreset-free PTAS designed specifically for the weighted $k$-means problem. In this paper, we present a simple PTAS for weighted $k$-means that does not rely on coresets. Building upon the framework of Jaiswal, Kumar, and Sen (2012) for the unweighted case, we extend the result to the weighted setting by using the weighted $D^2$-sampling technique. Our algorithm runs in time $n d \cdot 2^{O\left(\frac{k^2}{\epsilon}\right)}$ and outputs a set of $k$ centers whose total clustering cost is within a $(1 + \epsilon)$-factor of the optimal cost. As a key application of the weighted $k$-means, we obtain a PTAS for the sensor coverage problem, which can also be viewed as a continuous locational optimization problem. For this problem, the best-known result prior to our work was an $O(\log k)$-approximation by Deshpande (2014), whereas our algorithm guarantees a $(1 + \epsilon)$-approximation to the optimal coverage cost even before applying refinement steps like Lloyd desent.

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arXiv2024

Greedy BST on Permutation Initial Tree

The Greedy binary search tree (BST) algorithm, like the Splay tree, is a prominent candidate for the \emph{dynamic optimality conjecture}. While Greedy satisfies many desirable properties of BST, its cost and analysis to execute a search sequence $S$ is known to depend heavily on the choice of the \emph{initial tree} configuration. Most prior analyses assume a flat (empty) initial tree, under which several tight bounds are established. In this work, we introduce the notion of a \emph{permutation initial tree}, a specific class of non-flat initial tree and prove that for any permutation search sequence $S=(s_1,s_2,\dots, s_n)$, there exists a permutation initial tree $I_p$ such that the cost of Greedy on $I_p$ is same as its cost on the flat initial tree. As an application of our result, we show that the \emph{preorder traversal conjecture} holds for Greedy when the initial tree is a permutation initial tree. While it was previously known that Greedy achieves an $O(n)$ cost on preorder sequences for flat initial tree (Chalermsook et al., FOCS 2015), our result demonstrates that the same linear bound holds when the initial tree is a permutation initial tree. This result also matches the $O(n)$ bound for Splay tree on preorder sequence when the initial tree aligns with the traversal order (Chaudhuri and H\"oft, SIGACT 1993).

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arXiv2023

The Group Access Bounds for Binary Search Trees

The access lemma (Sleator and Tarjan, JACM 1985) is a property of binary search trees that implies interesting consequences such as static optimality, static finger, and working set property. However, there are known corollaries of the dynamic optimality that cannot be derived via the access lemma, such as the dynamic finger, and any $o(\log n)$-competitive ratio to the optimal BST where $n$ is the number of keys. In this paper, we introduce the group access bound that can be defined with respect to a reference group access tree. Group access bounds generalize the access lemma and imply properties that are far stronger than those implied by the access lemma. For each of the following results, there is a group access tree whose group access bound Is $O(\sqrt{\log n})$-competitive to the optimal BST. Achieves the $k$-finger bound with an additive term of $O(m \log k \log \log n)$ (randomized) when the reference tree is an almost complete binary tree. Satisfies the unified bound with an additive term of $O(m \log \log n)$. Matches the unified bound with a time window $k$ with an additive term of $O(m \log k \log \log n)$ (randomized). Furthermore, we prove simulation theorem: For every group access tree, there is an online BST algorithm that is $O(1)$-competitive with its group access bound. In particular, any new group access bound will automatically imply a new BST algorithm achieving the same bound. Thereby, we obtain an improved $k$-finger bound (reference tree is an almost complete binary tree), an improved unified bound with a time window $k$, and matching the best-known bound for Unified bound in the BST model. Since any dynamically optimal BST must achieve the group access bounds, we believe our results provide a new direction towards proving $o(\log n)$-competitiveness of Splay tree and Greedy.

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arXiv2022

Improved Pattern-Avoidance Bounds for Greedy BSTs via Matrix Decomposition

Greedy BST (or simply Greedy) is an online self-adjusting binary search tree defined in the geometric view ([Lucas, 1988; Munro, 2000; Demaine, Harmon, Iacono, Kane, Patrascu, SODA 2009). Along with Splay trees (Sleator, Tarjan 1985), Greedy is considered the most promising candidate for being dynamically optimal, i.e., starting with any initial tree, their access costs on any sequence is conjectured to be within $O(1)$ factor of the offline optimal. However, in the past four decades, the question has remained elusive even for highly restricted input. In this paper, we prove new bounds on the cost of Greedy in the ''pattern avoidance'' regime. Our new results include: The (preorder) traversal conjecture for Greedy holds up to a factor of $O(2^{α(n)})$, improving upon the bound of $2^{α(n)^{O(1)}}$ in (Chalermsook et al., FOCS 2015). This is the best known bound obtained by any online BSTs. We settle the postorder traversal conjecture for Greedy. The deque conjecture for Greedy holds up to a factor of $O(α(n))$, improving upon the bound $2^{O(α(n))}$ in (Chalermsook, et al., WADS 2015). The split conjecture holds for Greedy up to a factor of $O(2^{α(n)})$. Key to all these results is to partition (based on the input structures) the execution log of Greedy into several simpler-to-analyze subsets for which classical forbidden submatrix bounds can be leveraged. Finally, we show the applicability of this technique to handle a class of increasingly complex pattern-avoiding input sequences, called $k$-increasing sequences. As a bonus, we discover a new class of permutation matrices whose extremal bounds are polynomially bounded. This gives a partial progress on an open question by Jacob Fox (2013).

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