minimalism and complexity in a sensor-rich worldsuri/psdir/patras.pdf · 2011-06-08 · minimalism...

Minimalism and Complexityin a Sensor-rich World

Subhash Suri

Computer ScienceUniversity of California, Santa Barbara

June 17, 2011

Subhash Suri (UCSB) Minimalism and Complexity June 17, 2011 1 / 47

Sensors → Data → Information

Impressive progress (hardware, protocols, lightweight computing).

Interdisciplinary research (energy, communication and computing).

First community to emphasize energy (Green computing).

This Talk: complexity of building applications on the sensed data.

Preliminary theory: more questions than answers.

Surprises: complexity and minimalism where none suspected.


Smart Roads: Complexity

Traffic bottlenecks cost billions of dollars in productivity and waste.

Sensor-driven intelligence for road networks.

Computational complexity of time-dependent path planning.

Focus on planning, not on real-time rerouting.


Mapping and Exploration: Minimalism

Small robots with low cost sensing.

Cheapest possible sensing for a task.

How much sensing is enough?

Possibility and impossibility results.


Smart Roads: Data Collection

Loop detectors on roads sense average speed.

MTA data management system archive.


From Sensing to Smart Paths

Speed variability on road segments across time.

What can we do with this data?

What path-planning services can we provide?

How do we compute shortest-time routes?


Time-Aware Path Planning: Network Model

A directed graph G = (V,E).

The cost or delay c(u, v) is a known function of time (sensors).

Compact data representation: piecewise linear delay function.

LAX

WH

SB

SM

SB

WH

8 : 00

9 : 20

WH

LAX

9 : 20

9 : 50

SB

LAX

8 : 00

9 : 50

SB

LAX

8 : 00

9 : 40 Departure time at Santa Barbara

ArrivaltimeatLAX

8 : 00

9 : 50

9 : 40

The optimal path from s to d, and its cost, varies with time.


Path Planning Queries

Depart SB for LAX to catch a flight at 8 PM.

I Arrival time in LAX if leaving SB at 4 PM?

I To arrive at LAX at 6 PM, when to leave SB?

I To arrive at LAX between 5 and 6:30, best time to leave?

I From link delay functions to arrival time plots over node pairs.


Time Expansion Graphs

Network snapshots at regular intervals.

Make copies of the graph at each interval.

Path computation flows across the time-expanded network.

Drawbacks: Memory intensive, bad worst-case discontinuities.


Reasoning in Time-Varying Networks

Maintain single network copy, piecewise linear delay functions.

Arrival time, instead of delay, as the basic function.

Composition: arrival time of e becomes departure for f.

Functions compose along edges, undergo minimization at nodes.

LAX

WH

SB

SM

SB

WH

8 : 00

9 : 20

WH

LAX

9 : 20

9 : 50

SB

LAX

8 : 00

9 : 50

SB

LAX

8 : 00


ArrivaltimeatLAX

8 : 00

9 : 50

9 : 40


Fixed Start or Arrival Time: Easy

Earliest arrival time for a given departure timeI minor modification of Dijkstra’s algorithm.I A(v) = A(u) +Duv(A(u)).


Arrival Time Function

Latest departure time for a given arrival time at destinationI easy by time reversal.

Arrival time function for all departure times is less trivial.

Computed by operating on functions rather than scalars costs.

But consider some subtleties (oddities) first.


Continuity, Waiting Policies, FIFO

Arbitrary networks exhibit strange behavior [Orda and Rom]

I Waiting forbidden.F Shortest path with loops.F Infinite paths with finite delay, etc.

1 3

2

41

1

1+ (t− 5)2

22

I Discontinuities: D(t) =

{100 if t 6 101 if t > 10


Continuity, Waiting Policies, FIFO

Non-FIFO delay functions.

t

dik(t)

We will assume FIFO edges: reasonable for road traffic.

Observe that non-FIFO + Waiting = FIFO


Arrival Function Computation for all Departure Times

Repeatedly perform edge updates until stabilization (Bellman-Ford).

Parameters are functions instead of scalars:I B[s, i, j](t)← A[s, i](t) +Dij(A[s, i](t)), for each (i, j) ∈ EI A[s, j](t)← mini|(i,j)∈E B[s, i, j](t), for each j ∈ V

i jDi,j

A[s, i] A[s, i]

The basic BF essentially works, assuming blackbox access to

I function addition, composition, and minimization.


Composition and Minimization

LAX

WH

SB

SM

SB

WH

8 : 00

9 : 20

WH

LAX

9 : 20

9 : 50

SB

LAX

8 : 00

9 : 50

SBLAX

8 : 00


ArrivaltimeatLAX

8 : 00

9 : 50

9 : 40

A[s,d](t) is the (piecewise linear) arrival time function.

Orda-Rom show that if functional-operations are O(1), then thisalgorithm stabilizes after O(nm) steps.

But are intermediate arrival functions really O(1) complex?


Hidden Complexity

Even if individual edge functions have O(1) complexity, intermediatearrival functions A[s, i] grow in complexity:

Breakpoints introduced by minimization.

Departure time at Santa Barbara

ArrivaltimeatLAX

8 : 00

9 : 50

9 : 40

How many pieces can A[s,d] have in the worst-case?


Complexity Question

The road network has n nodes, m edges.

The delay function of each edge is piecewise linear with O(1) pieces.

Combinatorial complexity of the arrival time function A[s,d]?

An initial mistaken upper bound of O(nm) [Dean ’99], thenconjecture of super-polynomial size [Dean ’04].


Complexity Result

|A[s,d]| = nΘ(logn)

Lower bound even with linear costs.

Upper bound with polynomial number of pieces in edge cost functions.


Lower Bound

Connection to parametric shortest paths (PSP).

Edge costs are linear functions of a parameter λ (load):

c(ei) = aiλ+ bi

The parametric cost of a path P = (e1, . . . , ek) is∑ki=1(aiλ+ bi).

I The costs simply add, while in TDSP they compose.

PSP has complexity nΘ(logn) [Carstensen ’84, Mulmuley-Shah ’00]

Our proof shows a transformation from the lower bound constructionof PSP to one for the TDSP.


Upper Bound

Sufficient to bound the number of breakpoints in A[s,d]

Two kinds of breakpointsI Primitive: those of input edge cost functions, or their images.

I Minimization: those generated by the function minimization at nodes.Each minimization breakpoint corresponds to a time at which theshortest path changes.

LAX

WH

SB

SM

SB

WH

8 : 00

9 : 20

WH

LAX

9 : 20

9 : 50

SB

LAX

8 : 00

9 : 50

SB

LAX

8 : 00

9 : 40 Departure time at Santa BarbaraArrivaltimeatLAX

8 : 00

9 : 50

9 : 40


Bounding Minimization Breakpoints

Primitive Lemma: km primitive breakpoints in any A[s,d].

Convexity Lemma: A[s,d] is convex between adjacent primitivebreakpoint images.

LAX

WH

SB

SM

SB

WH

8 : 00

9 : 20

WH

LAX

9 : 20

9 : 50

SBLAX

8 : 00

9 : 50

SB

LAX

8 : 00


ArrivaltimeatLAX

8 : 00

9 : 50

9 : 40


Upper Bound for Linear Cost Functions

|Alin[s,d] 61

2(2n+ 1)logn+1

Convert to a layered graph.

v1v2

v3

v4

v5

v6

v1 v6

v2

v3

v4

v5

v2

v3

v4

v5

v2

v3

v4

v5

v2

v3

v4

v5


Upper Bound for Linear Cost Functions

Gn,c layered graph with n = 2p − 1 columns, c vertices per column.

Induction on p that |Alin[s,d]| 6 (n+1)2 clog(n+1).

s d

u

2p − 1 2p − 1M

c

Base case p = n = 1.I At most c paths, and minima of c lines has at most c breakpoints.

For a u in the middle column, |A[s,d|u]| 6 |A[s,u]|+ |A[u,d]|.


Results and Open Questions

A[s,d] can be super-polynomial size in the worst-case.

The complexity is not bad in simulations.

Under what conditions is A[s,d] provably well-behaved?

I A[s,d] linear if slopes in {0,α−β,α−β+1, . . . ,αβ}.

I Does planarity of road networks help?I What are natural parameters of road networks and delay functions.


Open Questions

Time-expansion graph blows up memory.

Time-based path sampling slows down queries.

Arrival Time function promising, but worst-case unappealing.

Good approximation of Arrival Time function?

What is the right, scalable, sound approach for time-dependent paths?


Minimalist Mapping and Monitoring

Imagine life withoutI Coordinates, distances, angles, or labels.I Only sensory input simple binary feature detection.

What can we learn about an environment?

How do we navigate, locale each other?

What can we not do?


Sensing and Mobility

Small flying, crawling, hopping robots.

Sensing, computing, mobility and actuation.

What tasks can be done with minimal sensing?


Sensing Minimalism

Minimalist design abstraction.

Insights in the role of sensing: what sensing is essential for which task.

Economics: low cost, low form factors etc.

Broader applicability: localization (GPS) not available everywhere.

Robustness to noisy measurements.


The Future?

The spider robots in movie Minority Report (2002)


A Binary Sensing Abstraction

Environment a polygon (unknown # vertices and holes)

No global frame of reference, labels, or metric measurements.

The robots “view” from a vertex v is a cyclic order of visible vertices.

Binary bits encoding “walls” or “gaps”

Combinatorial visibility vector cvv = (1, 0, 1, 0, 1)



Wall following or LoS vertex-vertex movement

What can we learn from CVVs alone?

Ignore control, comm, coordination, consensus etc for now.



Robots take snapshots at vertices (corners) only.

Same cvv = (1, 1, 1, 1) in all three case below

A related but continuous gap sensing model by LaValle.

What can or cannot be learned from CVVs alone?


Local vs. Global

CVVs insufficient to resolve convexity of a particular vertex.

cvv of a and c are identical.

But it is possible to decide if the whole polygon is convex!


Global Convexity

Environment is convex iff every vertexs cvv is a vector of all 1s. Why?

Each polygon has at least one convex vertex.

If non-convex, a convex vertex v has a reflex neighbor

The cvv of v cannot be all 1s.

Global knowledge despite local uncertainty.


Referencing

The relative order: w is 3rd ccw vertex in cvv of v

But, when at w, robot cannot identify v.

Use another robot (or, pebble) to mark a location.

To traverse the boundary of P once, robot puts a pebble at somevertex, cycles along the boundary until returning to pebble.

This counts the polygon size (number of vertices).


Topology: Global from local

Are CVVs sufficient to decide if a polygon is simply-connected?

Does it have holes? How many?


Hole Discovery

Must be a hole vertex visible from vertex of outer boundary:I discrete sensing suffices.I Hole graph is connected.I Not true in 3D!

Idea: Leave pebble at a vertex. Move to a visible vertex, and cyclearound the boundary until

I encountering the pebble, orI deciding that pebble and robot are on different boundaries.


Hole Discovery Results

Theorem 1: A robot with a single pebble can decide the topologicalsimplicity of the polygon.

Theorem 2: A robot with k+1 pebbles can identify all k holes in thepolygon. The number of pebbles can be reduced to 1.

Open Question: Is the pebble necessary?


Local vs. Global

Robots can detect multiple boundaries

But cannot distinguish holes from outer boundary!

Seems equivalent to sensing global cyclic sense (cw or ccw)

So, rendezvous protocols of moving to outer boundary don’t work


Monitoring: Visibility Coverage

Can a group of robots self-deploy to cover a polygon?

Art Gallery Theorem: A n-gon can be guarded by n/3 guards.

All proofs are geometric, using coordinates and vertex labels.Not clear why knowledge of coordinates essential.

Theorem: Guarding problem can be solved in the CVV model.



Visibility graph of a polygon.

VG recognition well-known open problem.

CVV sequence = Unlabeled VG

But by definition has a polygon associated

A CVV sequence fixes the number of pockets in each vertexs view,and recursively.

Does a CVV then also fix the corresponding VG?



Surprisingly, not!

CVV does not uniquely fix a polygon (in pairwise visibility sense).


Open Problems

Many search questions do not inherently require coordinates:I Target CountingI Search and RescueI Pursuit Evasion, Formation of patterns, etc.

Which of these and how well can be solved in CVV or some othersimple sensory model?


Thank You!


Acknowledgments

Algorithms Research GroupI L. Foschini, S. Gandhi, P. Kamousi, K. Klein and H. Yildiz

CollaboratorsI J. Hershberger (Mentor), P. Widmayer (ETH).I F. Bullo (UCSB), S. LaValle (UIUC), L. Guibas (Stanford), R.

Govindan (USC)

Sponsors (National Science Foundation)


minimalism and complexity in a sensor-rich worldsuri/psdir/patras.pdf · 2011-06-08 · minimalism...

Documents