Question

Statistically, the arrival of spring typically results in increased accidents

and increased need for emergency medical treatment, which often requires

blood transfusions. Consider the problem faced by a hospital that

is trying to evaluate whether its blood supply is sufficient.

The basic rule for blood donation is the following. A person’s own

blood supply has certain antigens present (we can think of antigens as a

kind of molecular signature); and a person cannot receive blood with a

particular antigen if their own blood does not have this antigen present.

Concretely, this principle underpins the division of blood into four types:

A, B, AB, and O. Blood of type A has the A antigen, blood of type B has the B

antigen, blood of type AB has both, and blood of type O has neither. Thus,

patients with type A can receive only blood types A or O in a transfusion,

patients with type B can receive only B or O, patients with type O can

receive only O, and patients with type AB can receive any of the four

types.4

(a) Let sO, sA, sB, and sAB denote the supply in whole units of the different

blood types on hand. Assume that the hospital knows the projected

demand for each blood type dO, dA, dB, and dAB for the coming week.

Give a polynomial-time algorithm to evaluate if the blood on hand

would suffice for the projected need.

This problem needs to be solved with network flow problem. As part of your solution,

you should draw a flow network. You should also discuss the eciency of your algorithm, i.e.

say what the eciency class is and why.

blood type supply demand

O 50 45

A 36 42

B 11 10

AB 8 3

Answer 1

The problem is a network flow scenario; using a flow network graph and a polynomial-time max-flow algorithm can determine if the hospital's blood supply meets the demand. Efficiency is polynomial-time, with O(V3) for Ford-Fulkerson or O(EV2) for Edmonds-Karp algorithms.

Step-by-step explanation:

The provided data suggests a network flow problem that can be addressed using a flow network to ensure a hospital's blood supply sufficiency. A network with nodes representing blood types (O, A, B, AB), supply and demand edges, and a polynomial-time algorithm like the max-flow min-cut theorem can determine if the current blood supply can meet projected demands. The nodes correspond to the different blood types, with directed edges representing possible transfusions following ABO transfusion protocols. The capacities of these edges match the number of units available (supply) or needed (demand).

Efficiency of Algorithm

The efficiency class of this network flow algorithm is typically O(V3) for the Ford-Fulkerson method or more efficient variations like the O(EV2) Edmonds-Karp algorithm. Since blood types follow specific compatibility rules, the graph's topology and the max-flow algorithm's polynomial time complexity make this approach both feasible and efficient for hospital blood supply management.