A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed number j \geq 3 of l's and each row contains a small fixed number k > j of l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixed j . When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixed j . A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. For j > 3 and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound.

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Low-Density Parity-Check Codes

During the last decade, sampling-based path planning algorithms, such as probabilistic roadmaps (PRM) and rapidly exploring random trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g. as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g. showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e. such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Sampling-based algorithms for optimal motion planning

The author addresses the problem of routing connections in a large-scale packet-switched network supporting multipoint communications. He gives a formal definition of several versions of the multipoint problem, including both static and dynamic versions. He looks at the Steiner tree problem as an example of the static problem and considers the experimental performance of two approximation algorithms for this problem. A weighted greedy algorithm is considered for a version of the dynamic problem which allows endpoints to come and go during the life of a connection. One of the static algorithms serves as a reference to measure the performance of the proposed weighted greedy algorithm in a series of experiments. >

Routing of multipoint connections

problems To understand the class of polynomial-time solvable problems, we must first have a formal notion of what a "problem" is. We define an abstract problem Q to be a binary relation on a set I of problem instances and a set S of problem solutions. For example, an instance for SHORTEST-PATH is a triple consisting of a graph and two vertices. A solution is a sequence of vertices in the graph, with perhaps the empty sequence denoting that no path exists. The problem SHORTEST-PATH itself is the relation that associates each instance of a graph and two vertices with a shortest path in the graph that connects the two vertices. Since shortest paths are not necessarily unique, a given problem instance may have more than one solution. This formulation of an abstract problem is more general than is required for our purposes. As we saw above, the theory of NP-completeness restricts attention to decision problems: those having a yes/no solution. In this case, we can view an abstract decision problem as a function that maps the instance set I to the solution set {0, 1}. For example, a decision problem related to SHORTEST-PATH is the problem PATH that we saw earlier. If i = G, u, v, k is an instance of the decision problem PATH, then PATH(i) = 1 (yes) if a shortest path from u to v has at most k edges, and PATH(i) = 0 (no) otherwise. Many abstract problems are not decision problems, but rather optimization problems, in which some value must be minimized or maximized. As we saw above, however, it is usually a simple matter to recast an optimization problem as a decision problem that is no harder. Encodings If a computer program is to solve an abstract problem, problem instances must be represented in a way that the program understands. An encoding of a set S of abstract objects is a mapping e from S to the set of binary strings. For example, we are all familiar with encoding the natural numbers N = {0, 1, 2, 3, 4,...} as the strings {0, 1, 10, 11, 100,...}. Using this encoding, e(17) = 10001. Anyone who has looked at computer representations of keyboard characters is familiar with either the ASCII or EBCDIC codes. In the ASCII code, the encoding of A is 1000001. Even a compound object can be encoded as a binary string by combining the representations of its constituent parts. Polygons, graphs, functions, ordered pairs, programs-all can be encoded as binary strings. Thus, a computer algorithm that "solves" some abstract decision problem actually takes an encoding of a problem instance as input. We call a problem whose instance set is the set of binary strings a concrete problem. We say that an algorithm solves a concrete problem in time O(T (n)) if, when it is provided a problem instance i of length n = |i|, the algorithm can produce the solution in O(T (n)) time. A concrete problem is polynomial-time solvable, therefore, if there exists an algorithm to solve it in time O(n) for some constant k. We can now formally define the complexity class P as the set of concrete decision problems that are polynomial-time solvable. We can use encodings to map abstract problems to concrete problems. Given an abstract decision problem Q mapping an instance set I to {0, 1}, an encoding e : I → {0, 1}* can be used to induce a related concrete decision problem, which we denote by e(Q). If the solution to an abstract-problem instance i I is Q(i) {0, 1}, then the solution to the concreteproblem instance e(i) {0, 1}* is also Q(i). As a technicality, there may be some binary strings that represent no meaningful abstract-problem instance. For convenience, we shall assume that any such string is mapped arbitrarily to 0. Thus, the concrete problem produces the same solutions as the abstract problem on binary-string instances that represent the encodings of abstract-problem instances. We would like to extend the definition of polynomial-time solvability from concrete problems to abstract problems by using encodings as the bridge, but we would like the definition to be independent of any particular encoding. That is, the efficiency of solving a problem should not depend on how the problem is encoded. Unfortunately, it depends quite heavily on the encoding. For example, suppose that an integer k is to be provided as the sole input to an algorithm, and suppose that the running time of the algorithm is Θ(k). If the integer k is provided in unary-a string of k 1's-then the running time of the algorithm is O(n) on length-n inputs, which is polynomial time. If we use the more natural binary representation of the integer k, however, then the input length is n = ⌊lg k⌋ + 1. In this case, the running time of the algorithm is Θ (k) = Θ(2), which is exponential in the size of the input. Thus, depending on the encoding, the algorithm runs in either polynomial or superpolynomial time. The encoding of an abstract problem is therefore quite important to our under-standing of polynomial time. We cannot really talk about solving an abstract problem without first specifying an encoding. Nevertheless, in practice, if we rule out "expensive" encodings such as unary ones, the actual encoding of a problem makes little difference to whether the problem can be solved in polynomial time. For example, representing integers in base 3 instead of binary has no effect on whether a problem is solvable in polynomial time, since an integer represented in base 3 can be converted to an integer represented in base 2 in polynomial time. We say that a function f : {0, 1}* → {0,1}* is polynomial-time computable if there exists a polynomial-time algorithm A that, given any input x {0, 1}*, produces as output f (x). For some set I of problem instances, we say that two encodings e1 and e2 are polynomially related if there exist two polynomial-time computable functions f12 and f21 such that for any i I , we have f12(e1(i)) = e2(i) and f21(e2(i)) = e1(i). That is, the encoding e2(i) can be computed from the encoding e1(i) by a polynomial-time algorithm, and vice versa. If two encodings e1 and e2 of an abstract problem are polynomially related, whether the problem is polynomial-time solvable or not is independent of which encoding we use, as the following lemma shows. Lemma 34.1 Let Q be an abstract decision problem on an instance set I , and let e1 and e2 be polynomially related encodings on I . Then, e1(Q) P if and only if e2(Q) P. Proof We need only prove the forward direction, since the backward direction is symmetric. Suppose, therefore, that e1(Q) can be solved in time O(nk) for some constant k. Further, suppose that for any problem instance i, the encoding e1(i) can be computed from the encoding e2(i) in time O(n) for some constant c, where n = |e2(i)|. To solve problem e2(Q), on input e2(i), we first compute e1(i) and then run the algorithm for e1(Q) on e1(i). How long does this take? The conversion of encodings takes time O(n), and therefore |e1(i)| = O(n), since the output of a serial computer cannot be longer than its running time. Solving the problem on e1(i) takes time O(|e1(i)|) = O(n), which is polynomial since both c and k are constants. Thus, whether an abstract problem has its instances encoded in binary or base 3 does not affect its "complexity," that is, whether it is polynomial-time solvable or not, but if instances are encoded in unary, its complexity may change. In order to be able to converse in an encoding-independent fashion, we shall generally assume that problem instances are encoded in any reasonable, concise fashion, unless we specifically say otherwise. To be precise, we shall assume that the encoding of an integer is polynomially related to its binary representation, and that the encoding of a finite set is polynomially related to its encoding as a list of its elements, enclosed in braces and separated by commas. (ASCII is one such encoding scheme.) With such a "standard" encoding in hand, we can derive reasonable encodings of other mathematical objects, such as tuples, graphs, and formulas. To denote the standard encoding of an object, we shall enclose the object in angle braces. Thus, G denotes the standard encoding of a graph G. As long as we implicitly use an encoding that is polynomially related to this standard encoding, we can talk directly about abstract problems without reference to any particular encoding, knowing that the choice of encoding has no effect on whether the abstract problem is polynomial-time solvable. Henceforth, we shall generally assume that all problem instances are binary strings encoded using the standard encoding, unless we explicitly specify the contrary. We shall also typically neglect the distinction between abstract and concrete problems. The reader should watch out for problems that arise in practice, however, in which a standard encoding is not obvious and the encoding does make a difference. A formal-language framework One of the convenient aspects of focusing on decision problems is that they make it easy to use the machinery of formal-language theory. It is worthwhile at this point to review some definitions from that theory. An alphabet Σ is a finite set of symbols. A language L over Σ is any set of strings made up of symbols from Σ. For example, if Σ = {0, 1}, the set L = {10, 11, 101, 111, 1011, 1101, 10001,...} is the language of binary representations of prime numbers. We denote the empty string by ε, and the empty language by Ø. The language of all strings over Σ is denoted Σ*. For example, if Σ = {0, 1}, then Σ* = {ε, 0, 1, 00, 01, 10, 11, 000,...} is the set of all binary strings. Every language L over Σ is a subset of Σ*. There are a variety of operations on languages. Set-theoretic operations, such as union and intersection, follow directly from the set-theoretic definitions. We define the complement of L by . The concatenation of two languages L1 and L2 is the language L = {x1x2 : x1 L1 and x2 L2}. The closure or Kleene star of a language L is the language L*= {ε} L L L ···, where Lk is the language obtained by

Introduction to Algorithms, Second Edition

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Sampling-based Algorithms for Optimal Motion Planning

A model of a burst-noise binary channel uses a Markov chain with two states G and B. In state G, transmission is error-free. In state B, the channel has only probability h of transmitting a digit correctly. For suitably small values of the probabilities, p, P of the B $rA G and G $rA B transitions, the model simulates burst-noise channels. Probability formulas relate the parameters p, P, h to easily measured statistics and provide run distributions for comparison with experimental measurements. The capacity C of the model channel exceeds the capacity C(sym. bin.) of a memoryless symmetric binary channel with the same error probability. However, the difference is slight for some values of h, p, P; then, time-division encoding schemes may be fairly efficient.

Capacity of a burst-noise channel

A Steiner minimal tree for given points $A_1 , \cdots ,A_n $ in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum lengt...

Steiner Minimal Trees

Random Plane Networks

Two channels are considered; a discrete channel which can transmit sequences of binary digits, and a continuous channel which can transmit band limited signals. The performance of a large number of simple signalling alphabets is computed and it is concluded that one cannot signal at rates near the channel capacity without using very complicated, alphabets.

A comparison of signalling alphabets

This paper gives a theoretical treatment of several properties which describe certain variable-length binary encodings of the sort which could be used for the storage or transmission of information. Some of these, such as the prefix and finite delay properties, deal with the time delay with which circuits can be built to decipher the encodings. The self-synchronizing property deals with the ability of the deciphering circuits to get in phase automatically with the enciphering circuits. Exhaustive encodings have the property that all possible sequences of binary digits can occur as messages. Alphabetical-order encodings are those for which the alphabetical order of the letters is preserved as the numerical order of the binary codes, and would be of possible value for sorting of data or consultation of files or dictionaries. Various theorems are proved about the relationships between these properties, and also about their relationship to the average number of binary digits used to encode each letter of the original message.

E. N. Gilbert

Papers

Capacity of a burst-noise channel

Steiner Minimal Trees

Random Plane Networks

A comparison of signalling alphabets

Variable-length binary encodings