Gavin Stewart and Benjamin Spitz underCharles Stoddard

Developing a Mathematical Model of

River Travel

Definitions

• k - The least number of possible next moves for a boat, unless the boat can reach the end of the river from an earlier campsite

• Y - The number of campsites on the river

Our Problem

• Based on the 2012 MCM problem

• Maximize the number of boats traveling down a river

• Constraints:o 6-18 day trip timeo 12 hours of travel in a day (8AM-8PM)o Only one boat at each campsite

• Known values:o 8 mph boat speedo 180 day boating seasono Y campsiteso River 225 miles long

Assumptions

• There exists a path from the start to the end of the river with each stop being at most 96 miles from the last.o Ensures that any boat can traverse the river

(96 is the number of miles a boat can travel in an optimal 12 hour traveling period).

• Except during the first and last six days, an equal number of boats are on the river at all times.o Conservation of flow allows for network

modeling techniques.

Generating Functions

• Take a sequence and change it into a polynomial

• Coefficients represent the presence of boatso 1 for occupiedo 0 for empty

• Exponent represents 0-indexed stop numbero Powers of x greater than or equal to Y

represent boats which have finished.

0n n

n

nxa a

Generating Functions as Schedules• Two operations

o Multiplying by x shifts boats over one stopo Adding one to a polynomial with a zero

constant term moves a boat from the start of the river to the first stop.

1 + 0x + 0x2 …

1 + 1x + 0x2 …

1 + 1x + 1x2 …

Day 1

Day 2

Day 3

Graph Theory

Multigraphs

• Have vertices

• Edges connect certain vertices

• Any number of edges can connect vertices.

Graphs

• Type of Multigraph

• Has only one edge between any two vertices.

Digraphs

• Directed Graph

• Type of Multigraph

• Each edge has a direction of travel.

The River as a Digraph

• Call the digraph representation of the river R.

• Beginning, end, and each stop are vertices.o Indices start at 0 and end at Y+1

• The edge (a,b) exists if a is at most 96 miles before b.

• The river is considered solvable if:o There is at least one path from the beginning

to the end.o

1( ) 1 ( ) { }Ydeg v v V R V ‚

Bottlenecks

• Two types:o River Configurationo Time Requirement

• The strictest bottleneck determines the flow of boats down the river.

Upper Bound When Ignoring Distance

• All conditions identical, except we assume a boat can travel from any stop to any other stop

• At most Y boats can finish the river in six dayso If more than Y boats finish, multiple boats

must occupy the same stop

Upper Bound When Ignoring Distance

• Use generating functionso Form:

o Initial Term: o First term moves boats on rivero Second term moves boats from start onto

river

1

1)()(

1

x

xxPP

nn

nn

00 P

11

1 2)(1)()(

xxx

x

x nnn

Upper Bound When Ignoring Time

• All constraints the same, except boats can finish in any number of days

• Let k be the minimum outdegree of a stop which has no stops before it connecting to the end of the river. o At most k boats can finish the river in a day.

• Approach:o Give each vertex of a river graph unit weight.o Construct a flow network with node splitting.o By max-flow min-cut, at most k boats can

travel down the river

Calculation of k

112 2 1 0

min{ ( )} 1u U

k deg u

1 1{ | ( ) { } ( , ) ( ) : ( , ) ( )}Y YU v v V R V u v E R u V E R ‚

Upper Bound When Ignoring Time

General Upper Bound

• Actual problem has same restrictions when ignoring time and distanceo Must have the same upper bounds as the two

less constrained problemso Max number of boats finishing in six days:

o Consider two types of rivers:

kY 6,max

kY 6

kY 6

• Travel time restriction stricter than flow restriction

• Max flow in six days is Y boatso First six days no boats finisho

• Stops spaced evenly enough to warrant generating function approach

YY 2916

180

Intuitive Development: Uniformly Spaced Stops• Stops divide river into segments

o Each segment miles long

o Boats travel stops per day

1Y

1

225

Y

)1(32

75

)1(96

225

YY

Intuitive Development: Uniformly Spaced Stops• Generating Function:

o

o

o Sends boats down when is not a multiple of six

o Sends more boats every sixth day when is not a multiple of six

( )( )

1

1

1

nn

n n

xxP P

x

1 cos ( 1)

3( ) 6

6 6 2

nY Y

n Y

6

Y

1n

Y

Intuitive Development: Uniformly Spaced Stops• Schedule is legal

o Boats take six days to finish Equality Always Holds

o Boats do not travel over 96 miles in a day Farthest traveled:

5

1

( ) 1 ( ) 1n

m m n Y

2255 966 1

YY

Y

39

43 375 326 2

YY Y

Intuitive Development:Uniformly Spaced Stops

• Schedule is Optimalo Y per six day periodo 29Y in a season

Non-Uniformly Spaced Stops• Previous schedule may not work

o May send out too many boats in a single day

• Spread out these boats over multiple days

66

0

1 ( ( 1 )) 1 ( ( 1))3 3( )

6 2 2

YY

t

cos n t cos nYn

( ) , 16 6

Y Yn

Non-Uniformly Spaced Stops

• Schedule Legalo Boats take 6 days to finish

Proof similar to the one for uniform• Summation of evaluates to Y

o Boats always travel no more than k stops When 6 divides k,

Otherwise,

( )6 6

Y Yk n

1 max ( )6 6

kY Y

n

Non-Uniformly Spaced Stops

• Schedule is optimalo In six days, Y boats travel down the river,

with the exception of the first six days 29Y boats finish

• Flow restriction stricter than travel time restriction

• Max flow in six days 6k boats

• Requires graph theory

• Divide river into subgraphs so exactly one boat can finish each day after the first sixo No vertices except the start and end shared

• The largest set of these river subgraphs contains exactly k subgraphs.o Cannot contain more than k; removing k vertices

makes the river unsolvable.o We can construct a set of size k indexed by r

Edges are in the subgraph whenever they are in

the parent graph

0 1, ( )Y rv v V H ( ) mod (0, 1)n rV H n r k Yv n

• Efficiencyo Boats can complete the river in six days

Boats travel over 96 miles in two days River is only 225 miles

o Boats do not finish in the first six days Boats finish for 174 of the 180 days

o 174k boats can finish in a season

Acknowledgements

• Charles Stoddard, Mentor and Resident Advisor

• Xcel Energy Foundation and 1984 Alumni, Sponsors

• The Kinder Morgan Foundation and the Spitz Family, Sponsors

• Nicholas Horianopoulos, Advisor and Instructor

• Krystal Meisel, Writing Advisor

• Lori Ball, FSI Program Director

• The Math and Science Teaching Institute (MAST)

• Maurice Irving Woods III and Anthony Gandara, Instructors

• Karen Allnutt, Resident Advisor

• Other FSI and MAST Staff

• Meghan Combs, poster assistance and co-mentee