The document presents a parallelized depth-first search (PDFS) algorithm for efficient environmental mapping by robot swarms. PDFS models the environment as a graph and divides the depth-first search exploration among multiple robots. At each time step, the algorithm assigns unfinished edges for robots to expand in parallel. While PDFS terminates when full exploration is complete, its worst-case runtime is no better than single-robot DFS due to scenarios where many robots finish early, leaving one robot as a bottleneck. The algorithm aims to efficiently map environments like crop fields to determine optimal pollination paths.

1. Parallelizing Depth-First Search for
Robotic Graph Exploration
PRESENTED BY,
VARUN NARANG
B.TECH, MATHEMATICS
AND COMPUTING
IIT GUWAHATI

2. Background
 The collective behavior of robot swarms fits them for
comprehensive search of an environment, which is
useful for terrain mapping, crop pollination, and
related applications.
 We study efficient mapping of an obstacle-ridden
environment by a collection of robots.

3. Abstract
 We model the environment as an undirected,
connected graph and require robots to explore the
graph to visit all nodes and traverse all edges.
 Here we present a graph exploration algorithm that
parallelizes depth-first search, so that each robot
completes a roughly equal part of exploration.

4. Motivation
 We consider the task of exploring a field of crops that
will subsequently be pollinated.
 Conducting exploration first is useful because once
the robot swarm has mapped the field entirely, it can
determine best paths to crops.
 The Harvard RoboBees project aims to build a
colony of small, autonomously flying robotic bees
that can accomplish pollination

5. Assumptions
 The environment is an undirected, connected graph
whose nodes represent locations in the environment.
 Edges represent the ability to move directly from one
location to another.
 An edge does not exist between adjacent locations if
an obstacle separates them, and does exist otherwise.
 A colony of homogeneous robots resides at a hive,
located at some node.

6.  An unlimited number of robots may occupy a node
simultaneously.
 At each time step, each robot starts at a node and
traverses an edge as instructed by the hive.
 We have implicitly assumed robots have perfect
odometry-they know their exact change in position at
every step. This implies robots always know their
coordinates relative to the hive.
 Exploration should terminate when the hive knows all
nodes and all edges. To this end, at each step all robots
transmit their locations back to the hive and notify the
hive of any crops detected.

7. Formal Model
 The environment is an undirected graph G=(V;E).
 An edge {v1, v2} ε E if no obstacle separates v1 and
v2.
 We require that G is connected so that all nodes are
reachable from vh.
 Based on the edges robots traverse and the nodes
where robots end up, the hive gradually builds a map
of G, denoted Gm = (Vm, Em). Every node visited for
the first time is added to Vm; every edge traversed
for the first time is added to Em.

8. Some Key Terms
 Nodes are labeled if each node contains information
that uniquely identifies it, and unlabeled otherwise.
 A graph has labeled nodes if a robot can recognize
a node it has visited previously.
 Undirected edges: The robot can move in either
direction on the edge then it is an undirected edge.

9. DFS algorithm
 Depth-first search, or DFS, is a way to traverse
the graph.
 Initially it allows visiting vertices of the graph only,
but there are hundreds of algorithms for graphs,
which are based on DFS.
 The principle of the algorithm is quite simple: to go
forward (in depth) while there is such possibility,
otherwise to backtrack.

11. Complexity analysis
 Assume that graph is connected.
 Depth-first search visits every vertex in the graph and
checks every edge its edge. Therefore, DFS complexity
is O(V + E).
 If an adjacency matrix is used for a graph representation,
then all edges, adjacent to a vertex can't be found
efficiently, that results in O(V2) complexity.
 DFS algorithm traverses undirected graphs with labeled
nodes, making 2*|E| edge traversals. Two along every
edge.

12.  Since any graph traversal requires at least |E| edge
traversals, and DFS requires 2|E| edge traversals, the
overhead of DFS is 2.
 For general graphs, even if the robot is given an
unlabeled map, an overhead of 2 is optimal; thus
DFS is a generally optimal algorithm with respect to
the authors' metric.

13. How to Reduce Overhead?
 Multiple robots should be able to divide the work of
exploration and reduce runtime more dramatically.
 Thus, our approach is to parallelize DFS.

14. Basic algorithm
 We initially provide three routines for the robot:
assign, expand, and backtrack.

15. DFS Algorithm
 Initially all vertices are white (unvisited).
 DFS starts in arbitrary vertex and runs as follows:
 Mark vertex u as gray (visited).
 For each edge (u, v), where u is white, run depth-first search
for u recursively.
 Mark vertex u as black and backtrack to the parent.

16. Each iteration of the while loop
represents one time step.
 At each time step, each robot
traverses at most one edge.
 A robot at a finished node cannot
expand, so it backtracks.
 A robot at a started but
unfinished node should expand
an edge.
 A robot at an un started node
should expand an edge if one
exists; otherwise it should
backtrack because the node is
already finished (the robot has
reached a dead end).

17.  The hive waits until the end
of the iteration to direct
expanding robots where to
go.
 It groups robots based on
their current node, then, for
each node with a robot, the
unexplored edges are
divided evenly among all
the robots there.
 Robots expand their
assigned edges, and the
hive adds these newly
explored edges to Em.

18. Correctness
Claim 1. PDFS terminates when G is completely
explored.
 Proof. Suppose G is completely explored. All nodes
have been finished, so the robots backtrack until
their stacks are empty. This is the termination
condition of PDFS.
 Now suppose PDFS terminates at time T.
 All nodes have been finished by time T.

19. Efficiency Analysis
 The runtime is the number of times the outer while
loop of PDFS iterates. We investigate the worst case
run time of the algorithm.
 For every k ≥ 1, ε< 1, we can construct a graph such
that PDFS has runtime at least 2|E|(1- ε ).

20. Proof (Efficiency)
 Consider the graph in Figure 3.1, consisting of a
“stem" of length r and a “trap" with B edges, where B
is high.
 The hive is the leftmost node of degree 3. The idea is
to use up all robots except one to explore the stem,
and leave the remaining robot to explore the trap.

21.  We have two cases:
 Case 1: k=3m
 At the i th iteration , 3m-i robots make it i-nodes down the stem, where there is unfinished work.
 In the first iteration, PDFS assigns 3(m-1) robots to each of the branches above and below the hive.
These robots arrive at finished nodes, backtrack to the hive and do nothing.
 At iteration m, a single robot makes it m nodes down the stem and enters the trap.
 All other robots fall into two categories: those with empty stacks, which are doing nothing, and
those whose stacks contain only finished nodes, which are backtracking to the hive.
 This graph contains three edges per unit length of the stem, so |E| = 3r +B.
 The runtime is the time taken by the robot that enters the trap: 2r to travel across the stem and
back, and 2B to explore the trap by single-robot DFS. We want 2r + 2B ≤ 2|E|(1-ε) for our claim to
hold.

22. We simply construct a graph with r = m and
appropriate B.
Case 2.k is not a power of 3.
We choose r = ┌ log3 (k), ┐making the stem long enough to
ensure only one robot enters the trap. We choose B according to
the equations above are satisfied.

23. Worst-Case Runtime
 The worst-case runtime of PDFS is 2|E|

24. Remarks
 PDFS performs no better than single-robot DFS in the
worst case.
 PDFS cannot take advantage of multiple robots when, in
a perverse graph, many robots finish their work, and
some robot becomes the bottleneck.
 We cannot allocate nished robots to unfinished work
because our requirement that there be no wasteful edge
traversals prevents them from leaving the hive after
finishing DFS.

25. Acknowledgements
 Parallelizing Depth-First Search for Robotic
Graph Exploration by Chelsea Zhang, Harvard
College, Cambridge, Massachusetts--April 1, 2010
 Distributed Graph Exploration- Gerard Tel--
April 1997
 Wikipedia 