Accelerated decomposition techniques for large discounted Markov decision processes
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Abstract
Many hierarchical techniques to solve large Markov decision processes (MDPs) are based on the partition of the state space into strongly connected components (SCCs) that can be classified into some levels. In each level, smaller problems named restricted MDPs are solved, and then these partial solutions are combined to obtain the global solution. In this paper, we first propose a novel algorithm, which is a variant of Tarjan’s algorithm that simultaneously finds the SCCs and their belonging levels. Second, a new definition of the restricted MDPs is presented to ameliorate some hierarchical solutions in discounted MDPs using value iteration (VI) algorithm based on a list of stateaction successors. Finally, a robotic motionplanning example and the experiment results are presented to illustrate the benefit of the proposed decomposition algorithms.
Keywords
Markov decision process Graph theory Tarjan’s algorithm Strongly connected components DecompositionIntroduction
The MDP theory is increasingly used in several problems of planning under uncertainty; it has proven tremendously useful in a wide area of disciplines (White 1993), including Communication Network, Games and various applications in Robotics. Other application areas of major importance, such as Precipitation Forecasting or Rainfall Estimations (Valipour 2016a, b), Evapotranspiration Estimations (Valipour et al. 2017; Rezaei and Valipour 2016), Water Lifting Devices (Yannopoulos et al. 2015) who need optimal decisionmaking are also fields of MDPs applications (Freier et al. 2011; Mishra and Singh 2011; Alighalehbabakhani et al. 2015).
Generally, these realworld problems have very large state spaces; it is impractical to solve them with the classical MDP algorithms, since their computational complexities are at least polynomials in the size of the state space (Littman et al. 1995).
Several studies have focused on tackling the curse of dimensionality: heuristic search (Bonet and Geffner 2003), action elimination techniques introduced by MacQueen (1967), decomposition techniques introduced by Ross and Varadarajan (1991) for constrained limiting average MDP and used by Abbad and Boustique (2003), Abbad and Daoui (2003), Daoui and Abbad (2007), Daoui et al. (2010) in several categories of MDPs (average, discounted and weighted MDPs. The weakness point of these decomposition methods is there polynomial runtime complexity. Dai and Goldsmith (2007) use also a decomposition technique named topological VI algorithm based on a linear Kosaraju–Sharir algorithm (Sharir 1981) but its disadvantage is to repeat, in each iteration, computing some constant terms. The parallelism is a known accelerated solution technique (Chen and Lu 2013). Chafik and Daoui (2015) combine the decomposition technique and parallelism, which leads also to a polynomial runtime decomposition algorithm.
Decomposing an MDP consists on: (1) partitioning the state space into SCCs and classifying these SCCs into some levels; (2) constructing and solving independently, in each level, smaller problems named restricted MDPs; (3) combining all the partial solutions to determine the global solution of the original MDP.
In this paper, discounted MDP with finite state and action space is considered. First, the authors present an accelerated version of the VI algorithm based on introducing for each stateaction pair a list of successors. They also propose another variant of Tarjan’s algorithm using an address table of the nodes.
The main contribution of this paper is a novel algorithm based on Tarjan’s algorithm that simultaneously finds the SCCs and classifies them into some levels. This algorithm accelerates the convergence time of the decomposition and so the hierarchical solutions in discounted reward MDP. Formerly, some accelerated hierarchical VI (AHVI) algorithms are proposed. The later AHVI algorithm is based on a new definition of the restricted MDPs.
Finally, a motivating robotic motionplanning example is presented to show the advantage of the proposed decomposition algorithms.
Markov decision processes
A Markov decision process models an agent, which interacts with its environment, taking as input the states of the environment and generating actions as outputs. MDPs are defined as controlled stochastic processes satisfying the Markov property and assigning reward values to state transitions (Bellman 1957; Puterman 1994).
Formally, an MDP is defined by the fivetuple (S, A, T, P, R), where S is the state space in which the process’s evolution takes place; A is the set of all possible actions which control the state dynamics; T is the set of time steps where decisions need to be made; P denotes the state transition probability function where \(P\left( {S_{t + 1} = jS_{t} = i,A_{t} = a} \right) = P_{iaj}\) is the probability of transitioning to a state j when an action a is executed in a state i, S _{ t } (A _{ t }) is a random variable indicating the state (action) at time t; −R provides the reward function defined on state transitions where R _{ ia } denotes the reward obtained if the action a is applied in state i.
A strategy \(\pi\) is defined by a sequence π = (π^{1}, π^{2}…) where \(\pi^{t} :H_{t} \to \varPsi\) is a decision rule, \(H_{t} = (S \times A)^{t  1} \times S\) is the set of all histories up to time t, and \(\varPsi = \{ (q_{1} , \ldots , q_{A} ) \in {\mathbb{R}}^{\left A \right} :\mathop \sum \nolimits_{i = 1}^{A} q_{i} = 1,\quad q_{i} \ge 0, \;1 \le i \le A \}\) is the set of probability distributions over \(A = \bigcup\nolimits_{i \in S} {A(i)}\).
A Markov strategy is a strategy π in which \(\pi^{t}\) depends only on the current state at time t, a stationary strategy is a Markov strategy with identical decision rules, and a deterministic or pure strategy is a stationary strategy whose single decision rule is nonrandomized. The core problem of MDPs is to find an optimal policy that specifies which action should be taken in each state.
Discounted reward MDPs
Let \(P_{\pi } (S_{t} = j,A_{t} = aS_{0} = i)\) be the conditional probability that at time t the system is in state j and the action taken is a, given that the initial state is i and the decision maker is a strategy π.
The objective is to determine \(V^{*}\), the maximum expected total discounted reward vector over an infinite horizon.
Accelerated value iteration algorithm
Value iteration algorithm is one of the most widely standard iterative methods used for finding optimal or approximately optimal policies in discounted MDPs. In this paragraph, the authors present an accelerated version of the VI algorithm (Algorithm 1) based on introducing for each action a the list of stateaction successors denoted by \(\varGamma_{a}^{ + }\), where \(\varGamma_{a}^{ + } \left( i \right) = \{ j \in S: P_{iaj} > 0\}\). Open image in new window
The Algorithm 1 permits to accelerate the iterations compared to the classical VI algorithm especially when the number of actions and successors is very less than the number of states. Indeed, the time complexity is reduced from \({\mathbf{\mathcal{O}}}\left( {\left A \right\left S \right^{2} } \right)\) to \({\mathbf{\mathcal{O}}}\left( {\left {\varGamma_{a}^{ + } } \right\left S \right} \right)\) per iteration where \(\left {\varGamma_{a}^{ + } } \right\) denotes the average number of stateaction successors.
Decomposition technique
Restricted MDPs

State space \(S_{pk} = C_{pk} \cup \bar{C}_{pk}\)
\(\bar{C}_{pk} = \{ j \in S,j \notin C_{pk} :\;\exists i \in C_{pk} , P_{iaj} > 0\}\)

Action space \(\begin{array}{*{20}l} {A_{pk} \left( i \right) = A\left( i \right)} \hfill & {{\text{if}}\;i \in C_{pk} } \hfill \\ {A_{pk} \left( i \right) = \theta } \hfill & {{\text{if}}\; i \in \bar{C}_{pk} } \hfill \\ \end{array}\)
\(\theta\) is some fictitious action that keeps the process in the same state where it is.

Transition \(\begin{array}{*{20}c} {P_{pk} \left( {ji,a} \right) = P_{iaj} } & {{\text{if}}\;i \in C_{pk} } \\ {P_{pk} \left( {ji,\theta } \right) = 1} & {{\text{if}}\;i \in \bar{C}_{pk} } \\ \end{array}\)

Reward function \(\begin{array}{*{20}c} {R_{pk} \left( {i,a} \right) = R_{ia} } & {{\text{if}}\;i \in C_{pk} } \\ {R_{pk} \left( {i,a} \right) = \left( {1  \alpha } \right)V^{*} \left( i \right)} & {{\text{if}}\;i \in \bar{C}_{pk} } \\ \end{array}\)
where \(V^{*} \left( i \right)\) is the optimal value calculated in some previous level.
Abbad and Daoui (2003) propose and show the correctness of the following decomposition algorithm (Algorithm 2) that finds an optimal strategy. Open image in new window
In the rest of this paper, the authors consider the decomposition algorithm and propose some optimizations to speed up these steps.
A variant of Tarjan’s algorithm
Tarjan’s algorithm (Tarjan 1972) is one of the most known approaches used for finding SCCs; it performs a depthfirst search (DFS) of the graph; it uses stack to push each visited node S _{0}, which is associated to the visited number S _{0}.idx assigned in the order in which nodes appear in the DFS. The ‘LowReturn’ number S _{0}.low is the smallest index of a node S_{i} in the stack reachable from the descendants of S _{0}, the SCC root is detected when (S _{0}.low = S _{0}.idx); the algorithm pop the stack one by one until the state popped is S _{0}. Tarjan’s algorithm requires \({\mathbf{\mathcal{O}}}\)(n + m) space and time where n is the number of nodes and m is the number of edges.
Dijkstra (1982) proposes a variant of Tarjan’s algorithm that maintains a stack of possible root S _{0} candidates instead of keeping track of lowreturn values; a SCC can be found by a second DFS starting at S _{0}. Couvreur (1999) designs a variant of Dijkstra’s algorithm for the purpose of finding accepting cycles that can be translated to SCCbased algorithm. Lowe (2014) proposes an iterative version for a multithreading, named concurrent algorithm.
In the initialisation step, the index value of each node (line 3 in Algorithm 3) is set to 0 to indicate that it is not visited; the index value is set to 2 when a node is first generated (line 5 in Algorithm 3); when a SCC is detected the index value of each node (line 22 in Algorithm 3) is set to 1 to indicate that it is in an SCC. Open image in new window
The search starts from the last node of the DLLV list (line 6 in Algorithm 3). Each visited node is moved to the end of the list (line 14 in Algorithm 3, Procedure 1). Open image in new window
Remark 1
In Tarjan’s algorithm, the step in line 4 requires \({\mathbf{\mathcal{O}}}\)(n) time, but in this variant version it only requires \({\mathbf{\mathcal{O}}}\)(k) time where k is generally a low constant compared to n, and depends on the structure of the graph. Eventually, the temporary complexity of Algorithm 3 remains similar to Tarjan’s algorithm (\({\mathbf{\mathcal{O}}}\)(n + m)), but the execution time is reduced.
Note that the first arbitrary element in DLLV list is useful to avoid testing if the node to be moved is in the beginning of the list or not.
Modified Tarjan’s algorithm for finding SCCs and levels
After the partition of the state space into SCCs, Abbad and Boustique (2003) use the following algorithm to classify these classes into some levels. Open image in new window
This algorithm requires 0(n ^{2}) time, to reduce the complexity of the decomposition algorithm (Algorithm 2) the authors propose a new algorithm that finds simultaneously the SCCs and theirs belonging levels. It is based on the following proprieties:
Let C be a SCC: \({\mathbb{L}}\left( C \right)\) design the level of C. For all x \(\in C\), \({\mathbb{L}}\left( x \right)\) design the level of x, and \({\mathbb{L}}\left( x \right) = {\mathbb{L}}\left( C \right)\).
Propriety 3.1
Let x and y be two nodes or states. If \(y \in \varGamma^{ + } \left( x \right)\) then \({\mathbb{L}}\left( x \right) \ge {\mathbb{L}}\left( y \right)\).
Proof
Let C _{ x } (C _{ y }) be the SCC containing x (y). Suppose that \({\mathbb{L}}\left( x \right) < {\mathbb{L}}\left( y \right)\) then \({\mathbb{L}}\left( {C_{x} } \right) < {\mathbb{L}}\left( {C_{y} } \right)\), from the definition of the levels, there is no successors of node x in \(C_{y}\), this contradicts that \(y \in \varGamma^{ + } \left( x \right)\).
Propriety 3.2
Let C be the SCC containing a node x and let y be a node such as \(y \notin C\), if \(y \in \varGamma^{ + } \left( x \right),\) then \({\mathbb{L}}\left( x \right) > {\mathbb{L}}\left( y \right)\), y is called external successors.
Proof
The Propriety 3.1 imply that \({\mathbb{L}}\left( x \right) \ge {\mathbb{L}}\left( y \right)\); \(y \in C\) imply that \({\mathbb{L}}\left( y \right) = {\mathbb{L}}\left( C \right)\) and \(x \notin C\) imply that \({\mathbb{L}}\left( x \right) \ne {\mathbb{L}}\left( C \right)\) and \({\mathbb{L}}\left( C \right) = {\mathbb{L}}\left( y \right)\) then \({\mathbb{L}}\left( x \right) > {\mathbb{L}}\left( y \right)\).
Propriety 3.3
Let \(C_{i}\), i = 1,…, p be the successor’s classes of C, \({\mathbb{L}}\left( C \right) = \left( {\max_{i} {\mathbb{L}}\left( {C_{i} } \right)} \right) + 1\); if (p = 0) then \({\mathbb{L}}\left( C \right) = 0\).
Proof
It is clear from the Propriety 3.2 and the definition of levels. Open image in new window
Theorem 1
The Algorithm 5 works correctly and runs in \({\mathbf{\mathcal{O}}}\)(n + m) time.
Proof
The proof follows from the Proprieties 3.1, 3.2 and 3.3. Indeed, the instructions of lines 19 and 23 transmit the SCC level to its root. In fact, after each recursive call, the function DFS_Levels(s) returns the possibly minimum level value to its predecessor. In this case, the level value is updated using the Propriety 3.1 (line 19). If any successor is external, the level value is updated using the Propriety 3.2 (line 23). When a SCC is detected, its level is determined by the level value transmitted to its root (Propriety 3.3). In this case (line 28 in Algorithm 5), the function should return the level value incremented by one (Propriety 3.2).
The Algorithm 5 has the similar structure as Tarjan’s algorithm so it runs in \({\mathbf{\mathcal{O}}}\)(n + m) time.
Remark 2
These properties can be easily applied to other varieties of Tarjan’s algorithm to simultaneously find the SCCs and their belonging levels, such as Dijkstra’s algorithm (Dijkstra 1982), Couvreur’s algorithm (Couvreur 1999), and Sequential or Concurrent Tarjan’s algorithm proposed by Lowe (2014).
Using Algorithm 5, the authors present the following first version of accelerated hierarchical VI (AHVI) algorithm (Algorithm 6). Open image in new window
Remark 3
 (i)
The restricted MDPs in the same level L _{ p } are independent, so they can be solved in parallel.
 (ii)
The DFS used in Tarjan’s algorithm ensures the dependency order of the restricted MDPs. In nonparallel computing, the MDP can be solved without passing through levels (Algorithm 7). Open image in new window
Remark 4
New restricted MDPs

State space \(S_{pk} = C_{pk}\)

Action space \(A_{pk} \left( i \right) = A\left( i \right),\;i \in C_{pk}\)

Transition \(P_{pk} \left( {ji, \, a} \right) = P_{iaj} ,\;i \in C_{pk}\)
 Reward function$$R_{pk} \left( {i,a} \right) = R_{ia} + \alpha \mathop \sum \limits_{{j \in \bar{\varGamma }_{a}^{ + } \left( i \right)}} P_{iaj} V^{*} \left( j \right), i \in C_{pk}$$(6)
Theorem 2
Let \(V_{pk}^{*} \left( i \right), \quad i \in C_{pk}\), the optimal value in the restricted MDP _{ pk }, then \(V_{pk}^{*} \left( i \right)\) is equal to the optimal value \(V^{*} \left( i \right)\) in the original MDP.
Proof
Then \(V_{0k}^{*} \left( i \right) = V^{*} \left( i \right)\) for all \(i \in C_{0k}\).
Let p > 0 and suppose that the result is true for all levels preceding p. Now, we show that the result is still true for level p.
The Eq. 6 and the fact that C _{ pk } U {L _{0},…,L _{ p−1}} is closed imply that \(V_{pk}^{*} \left( i \right)\) is equal to the optimal value \(V^{*} \left( i \right)\) in the original MDP.
Remark 5
The new state space definition does not consider the external successors of each SCC compared to the old definition of restricted MDPs. This reduces the state space and so the time required.
The following procedure constructs the new restricted MDPs. Open image in new window
For each external successor j of stateaction (i, a) the resulted reward is added (line 5 in Procedure 3) and the external successor is eliminated (line 6 in Procedure 3).
Using this procedure, the following third version of AHVI algorithm for discounted MDPs is presented: Open image in new window
Remark 6
It is more interesting to construct the new restricted MDP during the first iteration of VI algorithm; indeed, the step 4 in Algorithm 8 can be executed in the first iteration of step 5. Also, an external successor can be detected with a different class identifier or a different level value.
Experimental results
The proposed algorithms are tested using Intel(R) Core(TM)2 Duo process (2.6 GHz), C++ implementation, Windows 7 operating system (32 bits) and random models (\(\varepsilon = 10^{  5} ,\; \alpha = 0.9,\; \varGamma_{a}^{ + }  = 20\)).
In the sequential Lowe’s variant algorithm, each node is pushed in two stacks and popped from two stacks, thus the time needed is longer than Tarjan’s algorithm.
In Dijkstra’s variant for SCC, instead of keeping track of lowreturn values that need \({\mathbf{\mathcal{O}}}\)(m _{1}) time, where m _{1} is the number of explored edges, it maintains a stack of possible root candidates and uses a second DFS for SCC members which need \({\mathbf{\mathcal{O}}}\)(m) time and generally m _{1} < m, where m is the number of edges, so the time is longer than Tarjan’s algorithm. But the memory consumption is reduced, since the lowreturn values are not used.
In the proposed variant, searching all SCCs (not only those reachable from one start state) need \({\mathbf{\mathcal{O}}}\)(k) additional time instead of \({\mathbf{\mathcal{O}}}\)(n) additional time in Tarjan’s algorithm and generally k < n. Figure 5 shows the reduction time using the proposed variant, and it is efficient since it directly construct the SCCs list for the hierarchical solution of the original MDP, eventually, it requires more memory since it uses doubly linked list.
Note that the number of classes and the size of each class affect the performance of the hierarchical algorithms. In addition, the time gained by the new definition of the restricted MDPs depends on the number of the external successors.
Problem example
In this section, the authors present an example of applying MDP decomposition in Robotics navigation where the environment structure can be decomposed into some regions. Each region corresponds to a SCC.
Model for Robotics navigation

Grid of the environment The grid method is used to model the environment, which is entirely discretized according to a regular grid.

States space Using the grid method, the state space is therefore a set of grids; each grid cell has an associated value stating, obstacle, free or goal state. The obstacle state can be eliminated from the state space.

Actions space The robot can be controlled through nine actions: the eight compass directions and the fictitious action \(\theta\) that keep the robot on the spot. The actions that move the robot to an obstacle are eliminated. In a goal state the possible action is \(\theta\).
Note that the average number of stateaction successors is very less than the number of states, so the proposed AVI algorithm (Algorithm 1) is efficient in this case.

Reward function The transition to a free state is characterized by a cost of energy equal to \(x\) when the action is diagonal, and \(\frac{x}{\sqrt 2 }\) when the action is horizontal or vertical. For any transition to goal state, the reward value is equal to the constant R _{ b } that is very higher than x.
 Transition function The transition function defines the uncertainty due to the effects of the actions; it is a problem data and can be determined by reinforcement learning. In this problem, the authors use the example of transition function indicated in Fig. 8.
The transition function is similar for the other actions.
Robot oriented to the nearest goal
In each desk exists a goal state (G), the black grid is an obstacle and the blue grids represent the border of the environment.
The fourth restricted MDPs: MDP_{ pk }, k = 0,…,3 in level p = 0 can be solved in parallel, thereafter the fifth restricted MDP: MDP_{ p0} in level p = 1 is solved.
The abstract goal state represents an external successor. It is eliminated during the construction of the new restricted MDP_{10}. In old definition of the restricted MDPs, it is considered as a goal state.
It can be seen from Figs. 12 and 13 that the optimal solution is similar to that obtained in Fig. 11 without using the decomposition technique. Thus, the original problem is decomposed into five smaller problems, which reduces the time complexity.
Conclusions
In this work, the authors have proposed a new algorithm that simultaneously finds the SCCs and classifies them into some levels used to accelerate the decomposition steps. The possible uses of this algorithm are certainly not limited to distributed or parallel solution for discounted MDPs. The authors have also proposed an accelerated version of HVI algorithm using a list of state–action successors and a new definition of the restricted MDPs; this allows us to reduce the time required. In future works, we try to combine the action elimination techniques with the proposed decomposition algorithms for more acceleration. The distributed solution and various areas of applications are also our perspectives.
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