reasoning with constraints def remove(self,elt)"""remove the element from the priority queue""if elt in self elt_mapself elt_map[elt][ self removed del self elt_map[elt def update_each_priority(self,update_dict)"""update values in the priority queue by subtracting the values in update_dict from the priority of those elements in priority queue ""for elt,incr in update_dict items()if incr ! newval self elt_map get(elt,[ ])[ incr assert newval < str(elt)+":"+str(newval+incr)+"-"+str(incrself remove(eltif newval ! self add(elt,newval def pop(self)"""removes and returns the (elt,valuepair with minimal value if the priority queue is emptyindexerror is raised ""self max_size max(self max_sizelen(self pq)keep statistics triple heapq heappop(self pqwhile triple[ =self removedtriple heapq heappop(self pqdel self elt_map[triple[ ]return triple[ ]triple[ eltvalue def top(self)"""returns the (elt,valuepair with minimal valuewithout removing it if the priority queue is emptyindexerror is raised ""self max_size max(self max_sizelen(self pq)keep statistics triple self pq[ while triple[ =self removedheapq heappop(self pqtriple self pq[ return triple[ ]triple[ eltvalue def empty(self)"""returns true iff the priority queue is empty""return all(triple[ =self removed for triple in self pqversion june