Statement

President Trample just had one of his tantrums! Thus, he ordered an air raid of P airplanes against his favourite enemy, the United Nations of Antarctica (UNA). The UNA consists of a single line of N houses, each with a certain height H_i. The P airplanes will follow this line one at a time, dropping bombs from house i=0 to N-1. Airplanes are not very precise, so they will only be able to hit the highest houses. More precisely, they only hit houses i whose height is strictly larger than that of both houses at its sides: H_i > H_i-1 and H_i > H_i+1. When a house is hit, however, it will be burned to the ground, and H_i will become zero from that moment on, possibly enabling the following planes to hit more houses (not the current plane). Giorgio, the newly elected emperor of the UNA, is rushing to calculate which houses will be hit, in order to help his fellow citizens to move to the safe houses. How many houses will be burned to the ground after all P airplanes have passed?