Estimation of Small s-t Reliabilities in Acyclic Networks

Estimation of Small - Reliabilities in Acyclic Networks

Marco Laumanns and Rico Zenklusen
{marco.laumanns,rico.zenklusen}@ifor.math.ethz.ch
ETH Zurich, Institute for Operations Research, 8092 Zurich, Switzerland
Abstract

In the classical - network reliability problem a fixed network is given including two designated vertices and (called terminals). The edges are subject to independent random failure, and the task is to compute the probability that and are connected in the resulting network, which is known to be -complete. In this paper we are interested in approximating the - reliability in case of a directed acyclic original network . We introduce and analyze an specialized version of a Monte-Carlo algorithm given by Karp and Luby. For the case of uniform edge failure probabilities, we give a worst-case bound on the number of samples that have to be drawn to obtain an approximation, being sharper than the original upper bound. We also derive a variance reduction of the estimator which reduces the expected number of iterations to perform to achieve the desired accuracy when applied in conjunction with different stopping rules. Initial computational results on two types of random networks (directed acyclic Delaunay graphs and a slightly modified version of a classical random graph) with up to one million vertices are presented. These results show the advantage of the introduced Monte-Carlo approach compared to direct simulation when small reliabilities have to be estimated and demonstrate its applicability on large-scale instances.

1 Introduction

In the classical - network reliability problem, a fixed graph with two special vertices and is given whose edges fail (disappear) independently of each other with some given probability. The task is to determine the probability that and are still connected in the resulting network after edge failures. A famous related problem is the all-terminal reliability problem, where the goal is to determine the probability that all vertices are still connected to each other after edge failures (for further information on various reliability problems see [1]). Both problems are known to be computationally hard (-complete) even for very restricted classes of graphs [16, 13]. In particular, the - reliability problem remains hard in the case when is a directed acyclic planar graph of maximal degree three [12]. Therefore we are interested in finding approximations.

Randomized algorithms respecting some relative error bound with high probability have shown to be interesting approaches for different reliability problems [9, 7, 8]. As we are interested in relative error bounds, there is a significant difference between the estimation of the failure probability of the network and estimating the probability that the network is intact after edge failures. This results from the fact that estimating small probabilities by sampling is in general much more difficult than estimating large probabilities. Moreover, the techniques used for the estimation of reliability values often differ significantly from those used for the estimation of the failure probability. Most of the literature concentrates on estimating the failure probability because in typical applications, as communication or electrical networks, we have a highly reliable network and are interested in accurately estimating the probability of rare failures.

In this paper, we concentrate on estimating the probability that the network is intact, i.e., the probability that there is a path from to after edge failures. This is motivated by certain models of spreading processes on networks, such as disease spreading, which can be mapped onto reliability problems [4, 14, 17, 10, 11]. In this context, a path from to represents the spread of a disease from to , and we are interested in estimating the probability of the rare event that the disease spreads over a long distance from to .

The basis of our approach is a method presented by Karp and Luby [9] for estimating the failure probability in an - reliability problem when the graph is planar, which can be easily adapted for estimating the probability of connectedness of and on an arbitrary graph . To be efficiently applicable, however, the underlying graph needs to fulfill some additional properties, such as very low intactness probabilities of the edges and low vertex degrees. Furthermore, a computationally expensive preprocessing phase is required, which computes different quantities needed to efficiently sample from the proposed sample space. This makes the method not suited for application on large networks. We show in this paper that in the case of a directed acyclic network , some of these problems can be circumvented, and the resulting algorithm can be applied to large-scale instances. The simplifications are due to the fact that generating an - path uniformly at random in an acyclic graph can be performed in linear time (see Section 5). Except for very restricted classes of initial networks such as series-parallel networks (see [15] for further information), no practically useful methods for estimating small - reliabilities on large networks are known. Considering directed acyclic graphs can thus be seen as a natural next step.

The paper is organized as follows. We begin with some preliminaries in Section 2. In Section 3 the direct Monte-Carlo approach is described and its efficiency is analyzed. Section 4 then presents our algorithm for the estimation of small reliabilities in directed acyclic graphs up to some technical details that are explained in Section 5. In Section 6 we discuss how many samples have to be drawn in our algorithm to obtain a good estimate of the reliability with high probability. Section 7 contains computational results on two types of random directed acyclic networks comparing the direct Monte-Carlo approach with our algorithm and demonstrating the applicability of our algorithm on large-scale instances.

2 Preliminaries

Let be a directed acyclic network, where

  • is the set of vertices (with ),

  • is the set of edges (with ), and

  • is a function associating a failure probability with every edge of the network.

We call a network satisfying these properties an acyclic reliability network. Furthermore, we fix two special vertices . Every edge fails with probability independently of the others. Let be the resulting graph over the intact edges after the realization of the edge failures. We say that is intact if it contains a path from to , otherwise we say that is in a failed state. Finally, the - reliability of the network is defined as the probability of being intact.

Let be the function which associates with every edge its probability of being intact, i.e., . It is sometimes easier to look at our reliability model in a different way where edges appear rather than disappear. In this model we would begin with an empty network and flip a biased coin for all potential edges to determine whether they appear.

We are mainly interested in  approximations for , i.e., algorithms returning an estimate of accurate to within a relative error of with probability at least . To determine how many samples we have to draw in a Monte-Carlo algorithm to obtain an  approximation we often refer to the Generalized Zero-One Estimator Theorem introduced by Dagum et al. in [2]. The theorem is repeated below.

Theorem 1 (Generalized Zero-One Estimator Theorem).

Let be a random variable taking values in and let denote independent random variables distributed identically to . If and then

We typically use the theorem in the following form. If then is an  approximation for (using the fact that a random variable taking values in satisfies ).

3 A direct Monte-Carlo approach

In this section we consider a simple Monte-Carlo approach and show that it is efficient for sufficiently large values of , but inefficient for the estimation of small reliabilities. In this approach we simply flip a biased coin for every edge and observe whether and are connected in the resulting graph. Let be the random variable corresponding to this approach where if the resulting network is intact and otherwise. The random variable has thus a Bernoulli distribution with parameter and the reliability is estimated without bias by generating independent realizations of and returning their empirical mean which we denote by . A central question when using this method is how large has to be chosen to obtain an  approximation. A direct application of Theorem 1 gives the following:

Theorem 2.

is an  approximation of if satisfies

When is bounded below by , is an FPRAS ( approximation algorithm with running time bounded by a polynomial in and the input size). The difficult case is the estimation of small values, i.e., small reliabilities . This problem motivated the construction of the algorithm to be presented next.

4 Monte-Carlo method for estimating small reliabilities

The backbone of our algorithm is an adaption of the Monte-Carlo method presented in [9]. Our algorithm exploits that, in a directed acyclic network, we can easily (in linear time) calculate the mean number of intact paths from to after the edge failures. This value is normally a good estimate for the reliability in highly unreliable networks as in such networks an intact state typically contains only a few paths from to . Using a Monte-Carlo approach, our algorithm then estimates the ratio between and the mean number of intact paths from to after edge failures. Multiplying this estimate with the mean number of intact paths from to yields finally an estimate for the - reliability of the network.

Note that the ratio between the mean number of intact paths from to after edges failures and the reliability to estimate, i.e., the reciprocal of the value we estimate in our algorithm, is exactly the mean number of intact paths from to after edge failures conditioned on the event that the network will be intact after the failure process. One of the main problems for the development of methods for directly estimating this ratio is the difficulty of choosing a sample out of the pool of intact states with probability proportional to its real appearance probability. In fact such a sampling procedure could easily be transformed into an FPRAS for the estimation of using techniques presented by Jerrum et al. [6].

4.1 Notation

Let be the set of all possible states of the network after the realization of the edge failures, where a state represents the collection of intact edges. Furthermore, let be the set of all intact states. For every state we denote by the probability that occurs after the edge failure process, i.e.,  . In particular, the weight of the set is the reliability we want to estimate, .

Let be the set of all paths in from to . A path is simply represented by a subset of the edges . For every state we denote by the set of all paths from to in state . A state is intact if and only if we have . With every path we associate a weight which is the probability that all edges on the path will be intact after the edge failure process,  .

The mean number of intact paths from to after edge failures can thus be written as . In Section 5 we will see how this quantity can be efficiently calculated in acyclic graphs.

4.2 Estimating the ratio between and the mean number of intact paths from to after edge failures

To estimate the ratio between the reliability and the mean number of intact paths from to after edge failures we sample out of the sample space

where we associate the weight with every element . This sample space has the following interesting properties. On the one hand, the weight of the sample space is exactly the mean number of intact paths from to after edge failures, i.e.,

On the other hand, it is easy to sample elements of with probability proportional to their weights by the following two-step procedure. In a first step, a path is drawn with probability proportional to its weight . How this can be done efficiently will be discussed in Section 5. In the second step, all edges not contained in are sampled corresponding to their appearance probabilities. The appeared edges of the second step together with the edges in form an intact state . The tuple is finally the sampled state.

We will now introduce influence values for the elements in such that the expected influence value of a sample of is equal to . The expected influence value can then be estimated by a standard Monte-Carlo algorithm.

The approach taken in [9] was to fix for every intact state an arbitrary - path . We begin by following this approach and introduce later on another idea to reduce the variance of the estimator. With every sample we associate an influence value which is equal to one if the sample is of the form and equal to zero otherwise. The influence value of a sample corresponds therefore to the realization of a Bernoulli variable with parameter

which is precisely the value we would like to estimate. By repeating the sampling procedure times and counting the fraction of samples of the form we therefore obtain an unbiased estimator for with variance

Another unbiased estimator for with smaller variance than can be obtained by associating an influence value with every sample , and we define to be simply the mean of the influence values of our samples. This approach is particularly interesting in our case as the fact of being acyclic allows to compute the value in linear time (see Section 5) and thus does not increase the overall worst-case complexity of our algorithm. The reduction of the variance decreases the expected number of iterations to perform for obtaining an approximation when applying a stopping criterion as presented in Section 6.

Unfortunately, we cannot guarantee some minimal decrease of the variance of the estimator compared to as there are instances of reliability networks where we have an arbitrarily low decrease. On the other hand, it is easy to find instances where the variance reduces by an arbitrarily high factor.

We finally estimate by multiplying by the weight of the sampling space () which can be calculated efficiently as shown in the next section. Algorithm 1 gives a pseudocode for an implementation of the - reliability estimation based on . Note that the sampling of the edges not on the path performed in Algorithm 1 in lines 1 to 1 can be done more efficiently as in general we do not need to sample all edges for calculating . In Section 5 we will see how this part of the algorithm can be improved and show how to perform the remaining unspecified parts of our algorithm. More precisely, we will discuss the following operations, where represents the time needed for generating a random number uniformly in :

  • Sampling in time a path according to line 1 of Algorithm 1,

  • Determining in time the weight of the sampling space used in line 1 of Algorithm 1,

  • Sampling edges not being on the initial path and calculating as needed in lines 1 to 1 of Algorithm 1.

Input : An acyclic reliability network with two special terminals and the number of iterations to perform
Output : An estimate for
;\Hy@raisedlink\hyper@anchorstartAlgoLine0.1\hyper@anchorend
for to do  ; \Hy@raisedlink\hyper@anchorstartAlgoLine0.2\hyper@anchorend
Draw a random path out of the set with probability proportional to ;\Hy@raisedlink\hyper@anchorstartAlgoLine0.3\hyper@anchorend
;\Hy@raisedlink\hyper@anchorstartAlgoLine0.4\hyper@anchorend
foreach do  Take a sample of a Bernoulli variable with parameter to determine whether the edge appears; \Hy@raisedlink\hyper@anchorstartAlgoLine0.5\hyper@anchorend
if (the edge appears) then  ; \Hy@raisedlink\hyper@anchorstartAlgoLine0.6\hyper@anchorend
end end Determine ;\Hy@raisedlink\hyper@anchorstartAlgoLine0.7\hyper@anchorend
;\Hy@raisedlink\hyper@anchorstartAlgoLine0.8\hyper@anchorend
end ;\Hy@raisedlink\hyper@anchorstartAlgoLine0.9\hyper@anchorend
Determine ;\Hy@raisedlink\hyper@anchorstartAlgoLine0.10\hyper@anchorend
return
Algorithm 1 Estimation of based on
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10

5 Algorithmic details

5.1 Sampling - paths

We begin by studying how paths can be efficiently sampled according to line 1 of Algorithm 1. The idea is to start at the terminal and then to construct a path to by successively adding new edges. The choice of a new edge augmenting the current partial path is done in the following way. With every edge we associate a weight which is the sum of the weights of all paths from to using edge , i.e.,

(1)

During the path sampling method, after a partial path from to some vertex is constructed, we choose an outgoing edge of vertex with probability proportional to the weights . It is easy to verify that this procedure effectively samples a path with probability proportional to as desired. Furthermore, the edge weights can be easily computed by the following procedure.

We suppose without loss of generality that every edge lies on at least one path from to in . All edges not satisfying this condition can be eliminated in time in a preprocessing step. We then determine a topological order of the vertices. By the condition mentioned above it is clear that will be the last vertex in the topological order. We go through the vertices in reverse topological order and determine at each step the weights of the edges entering the current vertex. At the first step we look at every edge incident to and determine its weight which is equal to . The weights of the other edges can then be determined in linear time by using the following recursive formula which follows from the definition of the weights in (1),

(2)

where for , (resp. ) denotes the set of all edges going out of (resp. all edges entering ).

5.2 Determining

Another subproblem in Algorithm 1 is the calculation of , the weight of the sample space. Having already calculated the weight function , this problem can easily be solved by expressing in terms of as follows.

5.3 Sampling edges outside the initial path and calculating

As the sampling of the remaining edges in lines 1 to 1 of Algorithm 1 is used only for the calculation of , we do not have to know all intact edges but only those on a path from to . One way of improving the procedure is to sample only edges that can be reached from . This can be done by keeping track of the set of nodes that are currently reachable from . At the beginning, these are the nodes on the initial path . Then all edges going out of will be sampled, and for every appeared edge we add its endpoint to . This procedure is repeated until there are no edges outgoing from that have not already been sampled.

Finally, the determination of in line 1 of Algorithm 1 can be easily done by an analogue technique to the one used to calculate where this time we work on the subgraph of over the edges in where we give a weight of one to every edge.

6 Number of samples to draw

In this section we analyze how many samples have to be drawn in our algorithm to obtain an approximation.

6.1 A priori bounds

Using Theorem 1 we can derive the following bound.

Theorem 3.

If the number of samples satisfies

then and are both approximations of .

In Subsection 6.3, we discuss upper bounds for the ratio which allow to apply Theorem 3 in practice and to formulate conditions a network has to satisfy under which Algorithm 1 is an FPRAS for estimating .

6.2 Stopping criteria

In practice, the use of a stopping criterion typically allows to reduce the number of samples to take as we can profit from information gained during the execution of the algorithm. Furthermore, we do not suffer from a possibly weak upper bound for . In [2], Dagum et al. present two stopping criteria, applicable to our algorithm and ensuring that the result of the algorithm is an approximation of . In the computational results, the second algorithm (named Approximation Algorithm ) is used.

6.3 Bounding the ratio

In this subsection, we discuss upper bounds on the ratio . Together with Theorem 3, these bounds allow us to bound the number of iterations needed for our algorithm to deliver an approximation. The bounds discussed in this section are correct not only for the case of directed acyclic reliability networks but also for the more general case of arbitrary directed, undirected or even mixed reliability networks.

Previous results

Karp an Luby [9] gave the following upper bound on the ratio (in a slightly different context).

Theorem 4.

When combining the above theorem with Theorem 3 we obtain that Algorithm 1 is an FPRAS if is bounded by a polynomial in the input size of the reliability network . In the special case of uniform edge failure probabilities, i.e., , Theorem 4 reduces to

(3)

and implies that if

(4)

then Algorithm 1 is an FPRAS for estimating .

Improved bound in the case of uniform edge failure probability

We now give a new bound on for the case of uniform edge failure probabilities, which is sharper than (3), especially in the case when the reliability network is not too dense and does not contain long paths from to .

To quantify the sparsity of a graph we introduce the notion of edge-vertex bound. We say that a graph has an edge-vertex bound of if for any subset of the vertices we have that there are at most edges in the subgraph of induced by the vertices . The best edge-vertex bound of a graph can be determined in polynomial time by reduction to a flow problem [3]. Furthermore, , where is the maximum degree of the vertices in , can be used as a simple valid edge-vertex bound. Our new bound is given by the following theorem (see Appendix for a proof).

Theorem 5.

Let be a reliability network with uniform edge failure probability , edge-vertex bound and let respectively be the minimal and maximal length of any - path in . Then we have

The first term of the minimum in our bound is a slight improvement over the bound given by (3). This can be seen by observing that . Contrary to bound (3), the first term of our bound may be sharp. Furthermore it is independent of the graph topology and can easily be generalized to non-uniform failure probabilities.

The second term of the minimum in Theorem 5 tries to exploit some structure of the underlying network and is particularly interesting for graphs with a low edge-vertex bound and without long paths. For example, when working with networks where is bounded by a constant and , Theorem 5 implies that Algorithm 1 is an FPRAS when , which is a much weaker condition than the bound given by (4).

7 Computational results

In order to test our algorithm we used two random generators for creating directed acyclic graphs with low reliability. These generators are introduced in the first part of this section. In a second part, we analyze the running time of our algorithm on networks created by these generators with different sizes and reliabilities. Furthermore, the proposed algorithm is compared to a direct Monte-Carlo simulation.

7.1 Test instances

Delaunay graphs (DEL)

Our generator for directed acyclic Delaunay graphs takes two parameters, the number of vertices and a uniform edge intactness probability . We begin by choosing points uniformly at random in the unit square and consider the undirected graph given by a Delaunay triangulation of these points. The two terminals and are chosen as two vertices with maximal Euclidian distance. We give a linear orientation to the edges corresponding to the vector from to , i.e., an (undirected) edge is oriented as if the vector from to and the one from to have a non-negative scalar product, otherwise we take the orientation . Finally, all edges get uniform intactness probability equal to . One can easily observe that this construction guarantees that every vertex lies on a path from to .

Topological construction (TC)

A second generator we use has three parameters, the number of vertices , a density parameter allowing to control the expected number of edges in the graph and a parameter influencing the intactness probabilities. We begin with an empty graph over vertices where and . The graph will be constructed such that is a topological order of the vertices. In a first step, all edges of the graph are introduced, then intactness probabilities are assigned to the edges.

For we introduce an edge from to . This ensures that all vertices are on a path from to . All other possible edges will be present with probability , i.e., for every with we add an edge with probability .

Finally, the intactness probability of an edge is a number chosen uniformly at random in the interval . By choosing , edges connecting topological near vertices have in general higher intactness probabilities than edges connecting vertices being far away from each other in the topological order. Therefore, smaller values for result in less reliable networks. The value of will typically be chosen in .

Parameter choice

The parameters were fixed in such a way that networks of different reliabilities were obtained for graphs with ,, and vertices. We generated DEL instances for every for every graph size . For the creation of TC instances the parameter was always chosen such that the expected degree of every vertex is equal to ten. This ensures that all graphs generated with the TC generator having the same number of vertices also have about the same number of edges and simplifies the comparison of running times. For every graph size instances were generated for .

The computational results presented in this section have been obtained on workstations equipped with an AMD processor 3200+ and 1GB of RAM.

7.2 Results and interpretations

As a first observation, we noticed that the dependence of the running time on and is essentially proportional to as predicted. We therefore fixed , for all results presented in this section.

Figure 1 shows the running time of the proposed algorithm as a function of the estimated reliability. As expected, the running time grows when larger reliabilities have to be estimated, as an intact state contains often several paths from to . TC instances with about the same reliability as DEL instances are much easier to tackle. This comes from the fact that DEL instances are rather locally connected, whereas most edges in TC instances were randomly chosen. Local connectedness has the effect that when having an intact path from to , there is a good chance that several small subpaths of can be replaced by other intact subpaths with the same start and endpoint. Every subpath which can be replaced in this way raises the number of intact paths from to . Furthermore, as the replaceable subpaths are small it is likely that large groups of them are disjoint, implying that various combinations of these subpath replacements yield new intact paths from to . Figure 1 shows that even instances with vertices could be solved in reasonable time as long as the estimated reliability was not too large.

Figure 1: Running times of the proposed algorithm in function of the estimated reliabilities for DEL instances (on the left) and TC instances (on the right).

For the comparison of the proposed Monte-Carlo algorithm with a direct Monte-Carlo approach, instances with vertices were used as most of these instances could have been solved in reasonable time by both algorithms. As well as the proposed Monte-Carlo algorithm, the direct Monte-Carlo algorithm was implemented by using the sampling technique explained in 5.3 allowing to reduce the time needed per iteration in most instances. Figure 2 shows the running times of both algorithms on DEL and TC instances with vertices. As expected, the direct Monte-Carlo approach has an approximately linear dependence on the reciprocal of the estimated reliability. Figure 2 shows the strength of the proposed algorithm when low reliabilities have to be estimated.

Figure 2: Running times of the proposed algorithm (prop. algo.) and the direct Monte-Carlo (MC) approach for DEL instances (on the left) and TC instances (on the right) with vertices.

As the running time of the direct Monte-Carlo approach depends nearly linear in (see Figure 2) and we expect that the proposed algorithm has a running time approximately proportional to (i.e. the reciprocal of the value to estimate), we expect the ratio between the running time of the proposed algorithm and the direct Monte-Carlo approach is approximately linear in . This is confirmed by Figure 3 showing the ratio of the running time of both algorithms in function of for DEL and TC instances with vertices. It is not surprising that both algorithms need about the same running time when is near to one. This observation allows to perform a simple a priori test for deciding which algorithm is better suited for a particular instance. Given an instance, we first calculate (in linear time). If it is likely that the proposed algorithm will be faster than the direct Monte-Carlo approach. When , the direct Monte-Carlo approach is likely to be the more efficient algorithm.

Figure 3: Comparison of the running time of the proposed algorithm and the direct Monte-Carlo approach in function of for networks with vertices.

8 Conclusions

An adapted version of the Monte-Carlo algorithm given by Karp and Luby in [9] was presented and analyzed. The new algorithm is specialized for directed acyclic graphs and is suited for the estimation of small reliabilities. Computational results show the successful application of the proposed algorithm on two types of randomly generated large-scale instances and its advantage compared to the direct Monte-Carlo approach when very small reliabilities have to be estimated. Previous algorithms for accurate estimation of - reliability were only applicable on either very small instances or on a very restricted class of initial networks. For the case of uniform edge failure probabilities, a worst-case bound on the number of samples to be drawn was given that sharpens a bound presented in [9] and is significantly stronger in the case of relatively sparse graphs without long paths from to .

One important open question in this domain is if there exists an FPRAS for estimating - reliability in directed acyclic graphs. It would be interesting to find algorithms allowing to tackle instances efficiently that cannot be solved in reasonable time by our algorithm or the direct Monte-Carlo approach. Another point is the generalization of the upper bound for for the case of non-uniform failure probabilities. Additionally for the case of general (not necessarily acyclic) networks there seem to be no practically efficient algorithms at the moment for the estimation of low reliabilities on large instances.

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Appendix

Proof of Theorem 5

Let be a sample drawn according to lines 1-1 of Algorithm 1. We define the influence value of the sample as in Section 4 as a random variable

As discussed in Section 4, the influence value of a sample from can be used as an unbiased estimator for as we have

By the above equation, the reciprocal of a lower bound on is an upper bound on . We will therefore deduce the bound given in Theorem 5 by deriving a lower bound bound on .

Let be the subset of all elements of the sample space where the initial chosen path has length and the total number of appeared edges is , i.e.,

It is clear that all elements appear with equal probability and can be sampled by the method described in Algorithm 2. This method is introduced just for theoretical analysis.

Choose uniformly a path with length ;\Hy@raisedlink\hyper@anchorstartAlgoLine0.1\hyper@anchorend
;\Hy@raisedlink\hyper@anchorstartAlgoLine0.2\hyper@anchorend
;\Hy@raisedlink\hyper@anchorstartAlgoLine0.3\hyper@anchorend
for to do  Choose an edge uniformly at random; \Hy@raisedlink\hyper@anchorstartAlgoLine0.4\hyper@anchorend
;\Hy@raisedlink\hyper@anchorstartAlgoLine0.5\hyper@anchorend
;\Hy@raisedlink\hyper@anchorstartAlgoLine0.6\hyper@anchorend
end return
Algorithm 2 Sampling uniformly from
6

6

6

6

6

6

With every edge added in the for-loop of Algorithm 2 we associate a multiplicity equal to if both endpoints of are saturated by edges in and equal to otherwise. Intuitively, the multiplicity is a measure for the influence of an added edge in Algorithm 2 on the ratio . The following lemma formalizes this intuition.

Lemma 1.

Let be an intact state constructed as described in Algorithm 2. Then we have

Before proving this lemma, we discuss the link between the lemma and Algorithm 2. In general there are multiple ways to get an intact state with Algorithm 2. Depending on how the state was obtained, the multiplicities associated with the edges in are different. As Lemma 1 is true for every possible way of obtaining state by Algorithm 2, it is applicable even to intact states that were not constructed through Algorithm 2. One just has to fix a possible way how the state could have been constructed by Algorithm 2, i.e., an - path has to be fixed as well as an order for the edges in specifying in which sequence those edges were chosen in Algorithm 2. The multiplicities can then be calculated with respect to this order, and Lemma 1 can be applied.

Proof of Lemma 1.

Let be an intact state constructed as described in Algorithm 2. We define to be the set of all edges added to the initial path during the construction of . Let be the following partitioning of the edges in .

We prove the following statement, which immediately implies Lemma 1.

Proposition.

For every set there exists at most one - path in the state that contains all edges of and none of .

Lemma 1 follows from the above proposition by the following observation. The proposition implies that there are at most as many different - paths in as there are subsets of . We therefore have

which finally implies

It remains to prove the proposition. Let and suppose that we have two different - paths such that both contain all edges of and none of . This implies that their symmetric difference contains a cycle consisting only of edges in , thus contradicting the fact that does not contain cycles (any cycle in contains at least one element of ).

With the aid of Lemma 1 we prove the following intermediate result.

Lemma 2.

Let with and be a random sample from the sample space according to Algorithm 1. We have

Proof of Lemma 2.

Let be a sample corresponding to a result of Algorithm 2 for the given and . Conditioned on , has therefore the same distribution as . Let be the random variables corresponding to the multiplicities of the edges in . By Lemma 1 we have

(5)

Let . Observe that independently of which path and which edges were chosen, we have that at most vertices in are saturated by the edges in ( vertices are saturated through and every additional edge saturates at most two new vertices in ). Let be the vertices in which are saturated by .

In Algorithm 2, the edge is chosen uniformly at random from the remaining edges . As is sparse with edge-vertex bound we have that at most edges have both endpoints in . Furthermore, of these edges were already chosen. We therefore have the following stochastic inequality (which is true for any realization of ):

The stochastic inequality above allows to give a simple bound on the following conditional expectation:

Applying the above inequality, the expectation in (5) can be developed as

The last inequality comes from the fact that for and for . Developing further, we finally get the result of Lemma 2.

Beginning with the result of Lemma 2 we now prove Theorem 5 by first weakening and then eliminating the conditioning on . Let be the set of all elements of the sample space where the initial chosen path has length , i.e.,

Let be the random variable corresponding to the number of edges that appeared additionally to the ones of the initial path, when drawing an element out of . Note that is binomially distributed as

Using Lemma 2 we get

(6)

By replacing by in the first term of the above maximum we get the first part of the inequality in Theorem 5, i.e.,

The remaining part of Theorem 5 will be shown by developing the second term of (6) further. We will use the inequality

which is true for any and a result shown by Hamza [5] stating that the distance between the median and the mean of a binomial random variable is at most . Therefore, when choosing , we have and get the following result:

This implies the second part of the inequality in Theorem 5 as

which completes the proof of Theorem 5.

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