Question

I am searching for a non-recursive depth first search algorithm on graphs
in Pascal (Delphi).

I need DFS for computing strongly or bi-connected components of large graphs.
Currently I am using a recursive variant of the algorithm: http://en.wikipedia.org/wiki/Tarjan%27s_strongly_connected_components_algorithm

The problem is that for such algorithm I must define a large amount of memory
to be used for a stack and that makes later problems in Windows 7,
where Open and Save Dialogs do not work because of several threads generated….

So again: I do not see how to rewrite the Tarjan DFS algorithm
to work without recursion. Do you have any suggestion –
or point to a non recursice algorithm for depth first search on graphs?

Thanks.

Answer 1

The algorithm as described on Wikipedia looks to reasonably easily be made non-recursive with an explicit stack. Starting out with that (included here for reference, in case Wikipedia changes):

Input: Graph G = (V, E)

index = 0                                         // DFS node number counter 
S = empty                                         // An empty stack of node
for all v in V do
  if (v.index is undefined)                       // Start a DFS at each node
    tarjan(v)                                     // we haven't visited yet

procedure tarjan(v)
  v.index = index                                 // Set the depth index for v
  v.lowlink = index                               // SHOULD BE v.lowlink = MAX_INT?
  index = index + 1
  S.push(v)                                       // Push v on the stack
  for all (v, v') in E do                         // Consider successors of v
    if (v'.index is undefined)                    // Was successor v' visited?
        tarjan(v')                                // Recurse
        v.lowlink = min(v.lowlink, v'.lowlink)
    else if (v' is in S)                          // Was successor v' in stack S? 
        v.lowlink = min(v.lowlink, v'.index )     // v' is in stack but it isn't in the dfs tree
  if (v.lowlink == v.index)                       // Is v the root of an SCC?
    print "SCC:"
    repeat
      v' = S.pop                                  
      print v'
    until (v' == v)

Step 1: Remove loops containing recursion, adding labels and gotos. This is necessary to make loop variables explicit, savable and restorable (needed during recursion-simulation with stacks). A label needs to be added after tarjan()‘s return, as we’ll jump to it in a moment.

procedure tarjan(v)
  v.index = index                                 // Set the depth index for v
  v.lowlink = index                               // SHOULD BE v.lowlink = MAX_INT?
  index = index + 1
  S.push(v)                                       // Push v on the stack
  succ = all (v, v') in E      // Consider successors of v
  succIndex = 0                // presume succ is 0-based
loop_top:
  if succIndex >= Length(succ) goto skip_loop
  v' = succ[succIndex]
  if (v'.index is undefined)                    // Was successor v' visited?
      tarjan(v')                                // Recurse
recursion_returned:
      v.lowlink = min(v.lowlink, v'.lowlink)
  else if (v' is in S)                          // Was successor v' in stack S? 
      v.lowlink = min(v.lowlink, v'.index )     // v' is in stack but it isn't in the dfs tree
  succIndex = succIndex + 1
  goto loop_top
skip_loop:
  if (v.lowlink == v.index)                       // Is v the root of an SCC?
    print "SCC:"
    repeat
      v' = S.pop                                  
      print v'
    until (v' == v)

Step 2: Introduce a stack which contains all the relevant state for storing our position and computation in the loop at any point where we may be returning from recursion, or starting out at the top of the loop.

The stack:

T = empty // T will be our stack, storing (v, v', succ, succIndex, state)

state is an enumeration (TopState, ReturnedState) encoding the location in the procedure. Here’s the procedure rewritten to use this stack and state rather than recursion:

procedure tarjan(v)
  while (T is not empty) do
    (v, v', succ, succIndex, state) = T.pop
    case state of
      TopState: goto top
      ReturnedState: goto recursion_returned
    end case
top:
    v.index = index                                 // Set the depth index for v
    v.lowlink = index                               // SHOULD BE v.lowlink = MAX_INT?
    index = index + 1
    S.push(v)                                       // Push v on the stack
    succ = all (v, v') in E      // Consider successors of v
    succIndex = 0                // presume succ is 0-based
loop_top:
    if succIndex >= Length(succ) goto skip_loop
    v' = succ[succIndex]
    if (v'.index is undefined)                    // Was successor v' visited?
      // instead of recursing, set up state for return and top and iterate
      T.push(v, v', succ, succIndex, ReturnedState) // this is where we return to
      T.push(v', empty, empty, empty, TopState) // but this is where we go first
      continue // continue the while loop at top
recursion_returned:
      v.lowlink = min(v.lowlink, v'.lowlink)
    else if (v' is in S)                          // Was successor v' in stack S? 
      v.lowlink = min(v.lowlink, v'.index )     // v' is in stack but it isn't in the dfs tree
    succIndex = succIndex + 1
    goto loop_top
skip_loop:
    if (v.lowlink == v.index)                       // Is v the root of an SCC?
    print "SCC:"
    repeat
      v' = S.pop                                  
      print v'
    until (v' == v)

Step 3: Finally, we need to make sure the entry conditions are correct, for the top-level code which calls tarjan. That can easily be done by an initial push:

procedure tarjan(v)
  T.push(v, empty, empty, empty, TopState)
  while (T is not empty) do
    (v, v', succ, succIndex, state) = T.pop
    case state of
      TopState: goto top
      ReturnedState: goto recursion_returned
    end case
top:
    v.index = index                                 // Set the depth index for v
    v.lowlink = index                               // SHOULD BE v.lowlink = MAX_INT?
    index = index + 1
    S.push(v)                                       // Push v on the stack
    succ = all (v, v') in E      // Consider successors of v
    succIndex = 0                // presume succ is 0-based
loop_top:
    if succIndex >= Length(succ) goto skip_loop
    v' = succ[succIndex]
    if (v'.index is undefined)                    // Was successor v' visited?
      // instead of recursing, set up state for return and top and iterate
      T.push(v, v', succ, succIndex, ReturnedState) // this is where we return to
      T.push(v', empty, empty, empty, TopState) // but this is where we go first
      continue // continue the while loop at top
recursion_returned:
      v.lowlink = min(v.lowlink, v'.lowlink)
    else if (v' is in S)                          // Was successor v' in stack S? 
      v.lowlink = min(v.lowlink, v'.index )     // v' is in stack but it isn't in the dfs tree
    succIndex = succIndex + 1
    goto loop_top
skip_loop:
    if (v.lowlink == v.index)                       // Is v the root of an SCC?
    print "SCC:"
    repeat
      v' = S.pop                                  
      print v'
    until (v' == v)

It could also be done by a jump, jumping immediately to top. The code can be further cleaned up, perhaps converted to use a while or repeat loop to eliminate some of the gotos, etc., but the above should be at least functionally equivalent, eliminating the explicit recursion.