Overriding operators in child class - c++

Question

Im writing a program that simulates a vacuum cleaning a room. There is an initial state of dirty spots and I want to use various AI algorithms to find the best paths to clean the room. By making the algorithms separate from the actual problem I think my solution will be very modular.
Each algorithm only knows about States. Each state can be expanded to children states. Here is my first algorithm, UniformCost:
#include<iostream>
#include<set>
class State {
public:
State(){}
bool operator< (const State& s) const;
bool isGoal();
std::set<State> expand();
};
class UniformCost {
private:
State startState;
std::set<State> closedList; //list of no repeated states
public:
State start;
void setStart(State s);
State* getSolution();
};
void UniformCost::setStart(State st) {
start = st;
}
State* UniformCost::getSolution() {
closedList.insert(start);
while(!closedList.empty()) {
State cur = *closedList.begin();
if(cur.isGoal()) {
return &cur;
}
closedList.erase(cur);
std::set<State> children = cur.expand();
for (std::set<State>::iterator i = children.begin(); i != children.end(); ++i) {
closedList.insert(*i);
}
}
}
My main application creates the initial Node that is a child class of State.
class Node : public State {
public:
std::pair<int,int> loc;
int g_val;
std::set<std::pair<int,int> > dirt;
std::vector<char> path;
bool isGoal() {
return dirt.size() == 0;
}
bool operator< (const State& s) const {
Node n = (Node) s;
if(loc == n.loc) {
return false;
}
if(g_val <= n.g_val) {
return true;
}
return false;
}
std::set<State> expand() {
std::set<State> ret;
return ret;
}
};
How can I override the operator in the Node class that is expecting a "operator< (const State&)"? Or a more general question, how would I handle future "casting" of States?