scholarly journals Analysis of Transmissions Scheduling with Packet Fragmentation

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Nir Namman ◽  
Raphaël Rom
Keyword(s):  
Approximation Algorithm ◽  
Bin Packing ◽  
Scheduling Problem ◽  
Worst Case ◽  
Average Case ◽  
Unit Capacity ◽  
Minimum Number ◽  
International Audience ◽  
The Cost ◽  
Item Fragmentation

International audience We investigate a scheduling problem in which packets, or datagrams, may be fragmented. While there are a few applications to scheduling with datagram fragmentation, our model of the problem is derived from a scheduling problem present in data over CATV networks. In the scheduling problem datagrams of variable lengths must be assigned (packed) into fixed length time slots. One of the capabilities of the system is the ability to break a datagram into several fragments. When a datagram is fragmented, extra bits are added to the original datagram to enable the reassembly of all the fragments. We convert the scheduling problem into the problem of bin packing with item fragmentation, which we define in the following way: we are asked to pack a list of items into a minimum number of unit capacity bins. Each item may be fragmented in which case overhead units are added to the size of every fragment. The cost associated with fragmentation renders the problem NP-hard, therefore an approximation algorithm is needed. We define a version of the well-known Next-Fit algorithm, capable of fragmenting items, and investigate its performance. We present both worst case and average case results and compare them to the case where fragmentation is not allowed.

scholarly journals Analysis of an Approximation Algorithm for Scheduling Independent Parallel Tasks

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Keqin Li
Keyword(s):  
Approximation Algorithm ◽  
Bin Packing ◽  
Random Variables ◽  
Packing Problem ◽  
Performance Ratio ◽  
Worst Case ◽  
Average Case ◽  
Parallel Tasks ◽  
International Audience ◽  
Worst Case Performance Ratio

International audience In this paper, we consider the problem of scheduling independent parallel tasks in parallel systems with identical processors. The problem is NP-hard, since it includes the bin packing problem as a special case when all tasks have unit execution time. We propose and analyze a simple approximation algorithm called H_m, where m is a positive integer. Algorithm H_m has a moderate asymptotic worst-case performance ratio in the range [4/3 ... 31/18] for all m≥ 6; but the algorithm has a small asymptotic worst-case performance ratio in the range [1+1/(r+1)..1+1/r], when task sizes do not exceed 1/r of the total available processors, where r>1 is an integer. Furthermore, we show that if the task sizes are independent, identically distributed (i.i.d.) uniform random variables, and task execution times are i.i.d. random variables with finite mean and variance, then the average-case performance ratio of algorithm H_m is no larger than 1.2898680..., and for an exponential distribution of task sizes, it does not exceed 1.2898305.... As demonstrated by our analytical as well as numerical results, the average-case performance ratio improves significantly when tasks request for smaller numbers of processors.

scholarly journals Stochastic Flips on Dimer Tilings

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Thomas Fernique ◽  
Damien Regnault
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Upper Bound ◽  
Numerical Experiments ◽  
Triangular Grid ◽  
Expected Number ◽  
Worst Case ◽  
Average Case ◽  
International Audience

International audience This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called $\textit{flips}$, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a $\Theta (n^2)$ bound, where $n$ is the number of tiles of the tiling. We prove a $O(n^{2.5})$ upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.

scholarly journals Optimal Computer Crash Performance Precaution

2012 ◽  
Vol Vol. 14 no. 1 (Distributed Computing and...) ◽  
Author(s):  
Efraim Laksman ◽  
Hakan Lennerstad ◽  
Lars Lundberg
Keyword(s):  
Completion Time ◽  
Optimal Allocation ◽  
Parallel Program ◽  
Parallel Computer ◽  
Upper And Lower Bounds ◽  
Worst Case ◽  
Worst Case Ratio ◽  
International Audience ◽  
Computer Crash ◽  
The Cost

Distributed Computing and Networking International audience For a parallel computer system with m identical computers, we study optimal performance precaution for one possible computer crash. We want to calculate the cost of crash precaution in the case of no crash. We thus define a tolerance level r meaning that we only tolerate that the completion time of a parallel program after a crash is at most a factor r + 1 larger than if we use optimal allocation on m - 1 computers. This is an r-dependent restriction of the set of allocations of a program. Then, what is the worst-case ratio of the optimal r-dependent completion time in the case of no crash and the unrestricted optimal completion time of the same parallel program? We denote the maximal ratio of completion times f(r, m) - i.e., the ratio for worst-case programs. In the paper we establish upper and lower bounds of the worst-case cost function f (r, m) and characterize worst-case programs.

scholarly journals On Compact Encoding of Pagenumber $k$ Graphs

2008 ◽  
Vol Vol. 10 no. 3 ◽  
Author(s):  
Cyril Gavoille ◽  
Nicolas Hanusse
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Time ◽  
Worst Case ◽  
Encoding Scheme ◽  
Information Theoretic ◽  
Auxiliary Table ◽  
Minimum Number ◽  
International Audience

International audience In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

scholarly journals Balanced Scheduling of School Bus Trips using a Perfect Matching Heuristic

2018 ◽  
Vol 2672 (48) ◽  
pp. 1-11 ◽  
Author(s):  
Ali Shafahi ◽  
Sanaz Aliari ◽  
Ali Haghani
Keyword(s):  
Heuristic Algorithm ◽  
School Setting ◽  
Mixed Integer ◽  
Scheduling Problem ◽  
School Bus ◽  
Heuristic Solution ◽  
Mip Model ◽  
Bus Scheduling ◽  
Minimum Number ◽  
The Cost

In the school bus scheduling problem, the main contributing factor to the cost is the number of buses needed for the operations. However, when subcontracting the pupils’ transportation, unbalanced tours can increase the costs significantly as the lengths of some tours can exceed the daily fixed driving goal and will result in over-hour charges. This paper proposes a mixed integer programming (MIP) model and a matching-based heuristic algorithm to solve the “balanced” school bus scheduling problem with fixed start times in a multi-school setting. The heuristic solution always has the minimum number of buses as it starts with a minimal number of tours and does not alter the number of tours during its balancing stage. The effectiveness of the heuristic is tested by comparing its solutions with results from solving the MIP using commercial solvers whenever solvers could find a good solution. To illustrate the performance of the MIP and the heuristic, 11 problems were examined with different numbers of trips which are all based on two real-world problems: a California case study with 54 trips and the Howard County Public School System with 994 trips. Our numerical results indicate the proposed heuristic algorithm can find reasonable solutions in a significantly shorter time. The balanced solutions of our algorithm can save up to 16% of school bus operation costs compared with the best solution found by solvers from optimizing the MIP model after 40 hours. The balancing stage of the heuristic decreases the standard deviation of the tour durations by up to 47%.

scholarly journals Lower bounds for sparse matrix vector multiplication on hypercubic networks

1998 ◽  
Vol Vol. 2 ◽  
Author(s):  
Giovanni Manzini
Keyword(s):  
Sparse Matrix ◽  
Sparse Matrices ◽  
Log P ◽  
Probability Measures ◽  
Worst Case ◽  
Average Case ◽  
Local Memory ◽  
Matrix Vector Multiplication ◽  
International Audience ◽  
Matrix Vector

International audience In this paper we consider the problem of computing on a local memory machine the product y = Ax,where A is a random n×n sparse matrix with Θ (n) nonzero elements. To study the average case communication cost of this problem, we introduce four different probability measures on the set of sparse matrices. We prove that on most local memory machines with p processors, this computation requires Ω ((n/p) \log p) time on the average. We prove that the same lower bound also holds, in the worst case, for matrices with only 2n or 3n nonzero elements.

scholarly journals Best Possible Approximation Algorithms for Single Machine Scheduling with Increasing Linear Maintenance Durations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Xuefei Shi ◽  
Dehua Xu
Keyword(s):  
Approximation Algorithm ◽  
Single Machine ◽  
Machine Scheduling ◽  
Single Machine Scheduling ◽  
Point Of View ◽  
Scheduling Problem ◽  
Worst Case ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Maintenance Activities

We consider a single machine scheduling problem with multiple maintenance activities, where the maintenance duration function is of the linear formft=a+btwitha≥0andb>1. We propose an approximation algorithm named FFD-LS2I with a worst-case bound of 2 for problem. We also show that there is no polynomial time approximation algorithm with a worst-case bound less than 2 for the problem withb≥0unlessP=NP, which implies that the FFD-LS2I algorithm is the best possible algorithm for the caseb>1and that the FFD-LS algorithm, which is proposed in the literature, is the best possible algorithm for the caseb≤1both from the worst-case bound point of view.

Approximation Algorithms for a New Truck Loading Problem in Urban Freight Transportation

2020 ◽  
Vol 54 (3) ◽  
pp. 690-702
Author(s):  
Jie Fan ◽  
Guoqing Wang ◽  
Matthias Thürer
Keyword(s):  
Approximation Algorithm ◽  
Bin Packing ◽  
Freight Transportation ◽  
Performance Ratio ◽  
Fixed Cost ◽  
Worst Case ◽  
Urban Freight ◽  
Truck Loading ◽  
Loading Problem ◽  
Urban Freight Transportation

Motivated by urban freight transportation practices in China, we study an optimal truck loading problem in which a fixed cost and an additional cost that depends on the number of unloading points are associated with each truck used. The truck loading problem is modeled as a one-dimensional bin packing problem, where the cost of each bin is a convex fixed-plus-linear function of the number of items in the bin. The objective is to minimize the total cost of bins used. We develop an asymptotic polynomial time approximation scheme and an efficient approximation algorithm with an asymptotic worst-case performance ratio of 1.5 to tackle the problem. Our computational experiments indicate that the approximation algorithm performs well for certain order patterns, and also reveal several important insights on using the freight charge scheme.

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