Competitive Online Algorithm Research Articles

The Online Graph Exploration Problem on Restricted Graphs

The purpose of the online graph exploration problem is to visit all the nodes of a given graph and come back to the starting node with the minimum total traverse cost. However, unlike the classical Traveling Salesperson Problem, information of the graph is given online. When an online algorithm (called a searcher) visits a node ν, then it learns information on nodes and edges adjacent to ν. The searcher must decide which node to visit next depending on partial and incomplete information of the graph that it has gained in its searching process. The goodness of the algorithm is evaluated by the competitive analysis. If input graphs to be explored are restricted to trees, the depth-first search always returns an optimal tour. However, if graphs have cycles, the problem is non-trivial. In this paper we consider two simple cases. First, we treat the problem on simple cycles. Recently, Asahiro et al. proved that there is a 1.5-competitive online algorithm, while no online algorithm can be (1.25-e)-competitive for any positive constant e. In this paper, we give an optimal online algorithm for this problem; namely, we give a $\frac{1+\sqrt{3}}{2}(\simeq1.366)$-competitive algorithm, and prove that there is no $(\frac{1+\sqrt{3}}{2}-\epsilon)$-competitive algorithm for any positive constant e. Furthermore, we consider the problem on unweighted graphs. We also give an optimal result; namely we give a 2-competitive algorithm and prove that there is no (2-e)-competitive online algorithm for any positive constant e.

An Experimental Study of New and Known Online Packet Buffering Algorithms

We present the first experimental study of online packet buffering algorithms for network switches. We consider a basic scenario in which m queues of size B have to be maintained so as to maximize the packet throughput. For this model various online algorithms with competitive factors ranging between 2 and 1.5 were developed in the literature. We first develop a new 2-competitive online algorithm, called HSFOD, which is especially designed to perform well under real-world conditions. In our experimental study we have implemented all the proposed algorithms, including HSFOD, and tested them on packet traces from benchmark libraries. We have evaluated the experimentally observed competitiveness, the running times, memory requirements and actual packet throughput of the strategies. The tests were executed for varying values of m and B as well as varying switch speeds. It shows that greedy-like strategies and HSFOD perform best in practice.

SIGACT news online algorithms column 13

This is a survey of last year's work on online competitive algorithms. The survey covers only conference papers, including those from SODA, FOCS, STOC, STACS, COCOON, ISAAC, ESA, ICALP, WAOA, and ISAAC. I tried to focus on research directly dealing with competitive analysis, and in many cases I had to make judgment calls, whether to include them or not. My initial intent was to cover only some highlights, representative of the current trends in the field, but somehow, over time, it grew to a much longer and comprehensive document. I feel compelled to add that, with the deadline approaching, I was not able to spend as much time on debugging as I should, and some errors and omissions are, I'm afraid, inevitable. If you spot anything that should be corrected, please let me know.

Bincoloring

We introduce a new problem that was motivated by a (more complicated) problem arising in a robotized assembly environment. The bin coloring problem is to pack unit size colored items into bins, such that the maximum number of different colors per bin is minimized. Each bin has size B ∈ N . The packing process is subject to the constraint that at any moment in time at most q ∈ N bins are partially filled. Moreover, bins may only be closed if they are filled completely. We settle the computational complexity of the problem and design an approximation algorithm for a natural version which gives a solution whose value is at most one greater than the optimal one. We also investigate the existence of competitive online algorithms, which must pack each item without knowledge of any future items. We prove an upper bound of 3 q − 1 and a lower bound of 2 q for the competitive ratio of a natural greedy-type algorithm, and show that surprisingly a trivial algorithm which uses only one open bin has a strictly better competitive ratio of 2 q − 1 . Moreover, we show that any deterministic algorithm has a competitive ratio Ω ( q ) and that randomization does not improve this lower bound even when the adversary is oblivious.

A New Algorithm for On-line Coloring Bipartite Graphs

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for $P_7$-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for $P_6$-free and $P_7$-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color $P_6$-free (or $P_7$-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.

Minimizing weighted flow time

We consider the problem of minimizing the total weighted flow time on a single machine with preemptions. We give an online algorithm that is O ( k )-competitive for k weight classes. This implies an O (log W )-competitive algorithm, where W is the maximum to minimum ratio of weights. This algorithm also implies an O (log n + log P )-approximation ratio for the problem, where P is the ratio of the maximum to minimum job size and n is the number of jobs. We also consider the nonclairvoyant setting where the size of a job is unknown upon its arrival and becomes known to the scheduler only when the job meets its service requirement. We consider the resource augmentation model, and give a (1 + ε)-speed, (1 +1/ε)-competitive online algorithm.

Energy-efficient algorithms for flow time minimization

We study scheduling problems in battery-operated computing devices, aiming at schedules with low total energy consumption. While most of the previous work has focused on finding feasible schedules in deadline-based settings, in this article we are interested in schedules that guarantee good response times. More specifically, our goal is to schedule a sequence of jobs on a variable-speed processor so as to minimize the total cost consisting of the energy consumption and the total flow time of all jobs. We first show that when the amount of work, for any job, may take an arbitrary value, then no online algorithm can achieve a constant competitive ratio. Therefore, most of the article is concerned with unit-size jobs. We devise a deterministic constant competitive online algorithm and show that the offline problem can be solved in polynomial time.

An Algorithmic View on OVSF Code Assignment

Orthogonal Variable Spreading Factor (OVSF) codes are used in UMTS to share the radio spectrum among several connections of possibly different bandwidth requirements. The combinatorial core of the OVSF code assignment problem is to assign some nodes of a complete binary tree of height h (the code tree) to n simultaneous connections, such that no two assigned nodes (codes) are on the same root-to-leaf path. A connection that uses a 2-d fraction of the total bandwidth requires some code at depth d in the tree, but this code assignment is allowed to change over time. Requests for connections that would exceed the total available bandwidth are rejected. We consider the one-step code assignment problem: Given an assignment, move the minimum number of codes to serve a new request. Minn and Siu propose the so-called DCA algorithm to solve the problem optimally. In contrast, we show that DCA does not always return an optimal solution, and that the problem is NP-hard. We give an exact nO(h)-time algorithm, and a polynomial-time greedy algorithm that achieves approximation ratio Θ(h). A more practically relevant version is the online code assignment problem, where future requests are not known in advance. Our objective is to minimize the overall number of code reassignments. We present a Θ(h)-competitive online algorithm, and show that no deterministic online algorithm can achieve a competitive ratio better than 1.5. We show that the greedy strategy (minimizing the number of reassignments in every step) is not better than Ω(h) competitive. We give a 2-resource augmented online algorithm that achieves an amortized constant number of (re-)assignments. Finally, we show that the problem is fixed-parameter tractable.

The Freeze-Tag Problem: How to Wake Up a Swarm of Robots

An optimization problem that naturally arises in the study of swarm robotics is the Freeze-Tag Problem (FTP) of how to awaken a set of "asleep" robots, by having an awakened robot move to their locations. Once a robot is awake, it can assist in awakening other slumbering robots. The objective is to have all robots awake as early as possible. While the FTP bears some resemblance to problems from areas in combinatorial optimization such as routing, broadcasting, scheduling, and covering, its algorithmic characteristics are surprisingly different. We consider both scenarios on graphs and in geometric environments. In graphs, robots sleep at vertices and there is a length function on the edges. Awake robots travel along edges, with time depending on edge length. For most scenarios, we consider the offline version of the problem, in which each awake robot knows the position of all other robots. We prove that the problem is NP-hard, even for the special case of star graphs. We also establish hardness of approximation, showing that it is NP-hard to obtain an approximation factor better than 5/3, even for graphs of bounded degree. These lower bounds are complemented with several positive algorithmic results, including: · We show that the natural greedy strategy on star graphs has a tight worst-case performance of 7/3 and give a polynomial-time approximation scheme (PTAS) for star graphs. · We give a simple O(log δ)-competitive online algorithm for graphs with maximum degree δ and locally bounded edge weights. · We give a PTAS, running in nearly linear time, for geometrically embedded instances.

Online competitive algorithms for maximizing weighted throughput of unit jobs

We study an online unit-job scheduling problem arising in buffer management. Each job is specified by its release time, deadline, and a nonnegative weight. Due to overloading conditions, some jobs have to be dropped. The goal is to maximize the total weight of scheduled jobs. We present several competitive online algorithms for various versions of unit-job scheduling, as well as some lower bounds on the competitive ratios.We first give a randomized algorithm RMix with competitive ratio of e/(e−1)≈1.582. This is the first algorithm for this problem with competitive ratio smaller than 2.Then we consider s-bounded instances, where the span of each job (deadline minus release time) is at most s. We give a 1.25-competitive randomized algorithm for 2-bounded instances, matching the known lower bound. We also give a deterministic algorithm Edfα, whose competitive ratio on s-bounded instances is 2−2/s+o(1/s). For 3-bounded instances its ratio is ϕ≈1.618, matching the known lower bound.In s-uniform instances, the span of each job is exactly s. We show that no randomized algorithm can be better than 1.25-competitive on s-uniform instances, if the span s is unbounded. For s=2, our proof gives a lower bound of 4−22≈1.172. Also, in the 2-uniform case, we prove a lower bound of 2≈1.414 for deterministic memoryless algorithms, matching a known upper bound.Finally, we investigate the multiprocessor case and give a 1/(1−(mm+1)m)-competitive algorithm for m processors. We also show improved lower bounds for the general and s-uniform cases.

Dynamic TCP Acknowledgment: Penalizing Long Delays

We study the problem of acknowledging a sequence of data packets that are sent across a TCP connection. Previous work on the problem has focused mostly on the objective function that minimizes the sum of the number of acknowledgments sent and on the delays incurred for all of the packets. Dooly, Goldman, and Scott presented a deterministic $2$-competitive online algorithm and showed that this is the best competitiveness of a deterministic strategy. Recently Karlin, Kenyon, and Randall developed a randomized online algorithm that achieves an optimal competitive ratio of $e/(e-1) \approx 1.58$. In this paper we investigate a new objective function that minimizes the sum of the number of acknowledgments sent and the maximum delay incurred for any of the packets. This function is especially interesting if a TCP connection is used for interactive data transfer between network nodes. The TCP acknowledgment problem with this new objective function is different in structure than the problem with the function considered previously. We develop a deterministic online algorithm that achieves a competitive ratio of $\pi^2/6 \approx 1.644$ and prove that no deterministic algorithm can have a smaller competitiveness. We also study a generalized objective function where delays are taken to the $p$th power for some positive integer $p$. Again we give tight upper and lower bounds on the best possible competitive ratio of deterministic online algorithms. The competitiveness is 1 plus an alternating sum of Riemann's zeta function and tends to 1.5 as $p\rightarrow \infty$. Finally, we consider randomized online algorithms and show that, for our first objective function, no randomized strategy can achieve a competitive ratio smaller than $3/(3 - 2/e)\approx 1.324$. For the generalized objective function we show a lower bound of $2/(2-1/e) \approx 1.225$.

The orthogonal CNN problem

The online CNN problem had no known competitive algorithms for a long time. Sitters, Stougie and de Paepe showed that there exists a competitive online algorithm for this problem. However, both their algorithm and analysis are quite complicated, and above all, their upper bound for the competitive ratio is 10 5. In this paper, we examine why this problem seems so difficult. To this end we introduce a nontrivial restriction, orthogonality, against this problem and show that it decreases the competitive ratio dramatically, down to at most 9.

Competitive on-line stream merging algorithms for media-on-demand

We consider the problem of minimizing the bandwidth needed by media-on-demand servers that use stream merging. We consider the on-line case where client requests are not known ahead of time. To facilitate stream merging, clients have the ability to receive data from two streams simultaneously and can buffer up to half of a full stream. We present a new family of on-line stream merging algorithms called dynamic tree algorithms. The bandwidth requirements of the best of these, the dynamic Fibonacci tree algorithms, are within a factor of the minimum between log φ ( n)+ O(1) and log φ (1/(2 D))+ O(1) from the off-line optimal, where n is the number of requests, D is the guaranteed maximum start-up delay measured as a fraction of the time for a full stream, and φ=(1+ 5 )/2 . The new on-line algorithms use a dynamic Fibonacci tree to control how new arrivals should merge with existing streams. Empirical studies show that the dynamic Fibonacci tree algorithms perform much better than indicated by the analysis.

Scheduling multicasts on unit-capacity trees and meshes

This paper studies the multicast routing and admission control problem on unit-capacity tree and mesh topologies in the throughput model. The problem is a generalization of the edge-disjoint paths problem and is NP-hard both on trees and meshes. We study both the offline and the online version of the problem: In the offline setting, we give the first constant-factor approximation algorithm for trees, and an O(( log log n) 2) -factor approximation algorithm for meshes. In the online setting, we give the first polylogarithmic competitive online algorithm for tree and mesh topologies. No polylogarithmic-competitive algorithm is possible on general network topologies (Lower bounds for on-line graph problems with application to on-line circuits and optical routing, in: Proceedings of the 28th ACM Symposium on Theory of Computing, 1996, pp. 531–540) and there exists a polylogarithmic lower bound on the competitive ratio of any online algorithm on tree topologies (Making commitments in the face of uncertainity: how to pick a winner almost every time, in: Proceedings of the 28th Annual ACM Symposium on Theory of Computing, 1996, pp. 519–530). We prove the same lower bound for meshes.

Competitive Call Control in Mobile Networks

In this paper we deal with competitive local on-line algorithms for non-preemptive channel allocation in mobile networks. The signal interferences in a network are modeled using an interference graph G . We prove that the greedy on-line algorithm is Δ -competitive, where Δ is the maximum degree of G . We employ the ``classify and randomly select" paradigm [5], [17], and give a 5 -competitive randomized algorithm for the case of planar interference graphs, a 2 -competitive randomized algorithm for trees, and a (2c) -competitive randomized algorithm for graphs of arboricity c . We also show that the problem of call control in mobile networks with multiple available frequencies reduces to the problem of call control in mobile networks with a single frequency. Using this reduction, we present on-line algorithms for general networks with a single frequency. We give a local on-line algorithm which is (α (δ +1 + α )/(1/2+α )2 )-competitive, where α is the independence number of G , and δ is the average degree of G . The above results hold in the case when the duration of each request is infinite, and the benefit the algorithm gains by accepting each request is equal to one. They are extended to handle requests of arbitrary durations, and arbitrary benefits.

Let caches decay

Power dissipation is increasingly important in CPUs ranging from those intended for mobile use, all the way up to high-performance processors for highend servers. Although the bulk of the power dissipated is dynamic switching power, leakage power is also beginning to be a concern. Chipmakers expect that in future chip generations, leakage's proportion of total chip power will increase significantly. This article examines methods for reducing leakage power within the cache memories of the CPU. Because caches comprise much of a CPU chip's area and transistor counts, they are reasonable targets for attacking leakage. We discuss policies and implementations for reducing cache leakage by invalidating and "turning off" cache lines when they hold data not likely to be reused. In particular, our approach is targeted at the generational nature of cache line usage. That is, cache lines typically have a flurry of frequent use when first brought into the cache, and then have a period of "dead time" before they are evicted. By devising effective, low-power ways of deducing dead time, our results show that in many cases we can reduce L1 cache leakage energy by 4x in SPEC2000 applications without having an impact on performance. Because our decay-based techniques have notions of competitive online algorithms at their roots, their energy usage can be theoretically bounded at within a factor of two of the optimal oracle-based policy. We also examine adaptive decay-based policies that make energy-minimizing policy choices on a per-application basis by choosing appropriate decay intervals individually for each cache line. Our proposed adaptive policies effectively reduce L1 cache leakage energy by 5x for the SPEC2000 with only negligible degradations in performance.

On-Line Competitive Algorithms for Call Admission in Optical Networks

We study the on-line call admission problem in optical networks. We present a general technique that allows us to reduce the problem of call admission and wavelength selection to the call admission problem. We then give randomized algorithms with logarithmic competitive ratios for specific topologies in switchless and reconfigurable optical networks. We conclude by considering full duplex communications.

Combining Fairness with Throughput: Online Routing with Multiple Objectives

This paper presents online algorithms for routing and bandwidth allocation which simultaneously approximate fair and max-throughput solutions. In fact, the algorithms solve a more difficult problem: for any bandwidth b, the number of sessions that get bandwidth b in the online algorithm is not smaller than the number of sessions receiving γb offline, where γ is the competitive ratio. This problem is provably harder than the problem of maximizing throughput (e.g., [4]) or the problem of maximizing the bandwidth assigned to the most starved session (e.g., [3]). For the case where the algorithm assigns bandwidths, we present an O(log2nlog1+εU/ε)-competitive algorithm, for any ε, where U is the minimum (over all choices of routes) of the maximum number of sessions routed along any single link. We also show an Ω(log1+εU/ε) lower bound in this model. For a more practically interesting model where the algorithm assigns routes and weights, and where these weights are used to drive the Weighted Fair Queuing policy in the routers, we present an O(log2nlogU)-competitive algorithm. We also show that the dependence on U is necessary by presenting an Ω(logU) lower bound. Previous upper and lower bounds for online maximization of throughput become invalid if we are allowed to assign weights. We prove an Ω(logn) lower bound for this model and present an O(lognloglogn)-competitive online algorithm. We present preliminary simulation results which show that our algorithm is effective in attaining high throughput without significantly sacrificing fairness.

Cache decay

Power dissipation is increasingly important in CPUs ranging from those intended for mobile use, all the way up to high-performance processors for high-end servers. While the bulk of the power dissipated is dynamic switching power, leakage power is also beginning to be a concern. Chipmakers expect that in future chip generations, leakage's proportion of total chip power will increase significantly. This paper examines methods for reducing leakage power within the cache memories of the CPU. Because caches comprise much of a CPU chip's area and transistor counts, they are reasonable targets for attacking leakage. We discuss policies and implementations for reducing cache leakage by invalidating and “turning off” cache lines when they hold data not likely to be reused. In particular, our approach is targeted at the generational nature of cache line usage. That is, cache lines typically have a flurry of frequent use when first brought into the cache, and then have a period of “dead time” before they are evicted. By devising effective, low-power ways of deducing dead time, our results show that in many cases we can reduce LI cache leakage energy by 4x in SPEC2000 applications without impacting performance. Because our decay-based techniques have notions of competitive on-line algorithms at their roots, their energy usage can be theoretically bounded at within a factor of two of the optimal oracle-based policy. We also examine adaptive decay-based policies that make energy-minimizing policy choices on a per-application basis by choosing appropriate decay intervals individually for each cache line. Our proposed adaptive policies effectively reduce LI cache leakage energy by 5x for the SPEC2000 with only negligible degradations in performance .

Greedy online algorithms for routing permanent virtual circuits

We analyze the competitive ratio of two greedy online algorithms for routing permanent virtual circuits in a network with arbitrary topology and uniform capacity links. We show that the competitive ratio of the first algorithm, with respect to network congestion, is in and where m is the number of links in the network, 𝒟 is the maximum ratio, over all requests, of the length of the longest path for the request to the length of the shortest path for the request, and ℒ︁ is the ratio of the maximum-to-minimum bandwidth requirement. We show that the competitive ratio of the second greedy algorithm is in Ω(d + log(n −d)) and min{O(d log n), when the optimal route assignment is pairwise edge disjoint, where n is the number of network nodes and d is the length of the longest path that can be assigned to a request. It is known that the optimal competitive ratio for this problem is Θ(log n). Aspnes et al. designed a Θ(log n) competitive online algorithm that computes an exponential function of current congestion to make each decision. The greedy online algorithms, although not optimal, make each decision more quickly and still have good competitive ratios in many nontrivial situations. © 1999 John Wiley & Sons, Inc. Networks 34: 136–153, 1999